Ab initio quantum molecular dynamics methodbased on the restricted-path integral: Application

to electron plasma and an alkali metal

Item Type text; Dissertation-Reproduction (electronic)

Authors Oh, Ki-Dong

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.

Download date 24/02/2021 08:54:29

Link to Item http://hdl.handle.net/10150/284195

INFORMATION TO USERS

This manuscript has been reproduced from the microfilm master. UMI

films the text directly fix)m the original or copy submitted. Thus, some

thesis and dissertation copies are in typewriter &ce, while others may be

fi-om any type of computer printer.

The quality of this reproduction is dependent upon the quality of the

copy submitted. Broken or indistinct print, colored or poor quality

illustrations and photographs, print bleedthrough, substandard margins,

and improper aligmnent can adversely affect reproduction.

In the unlikely event that the author did not send UMI a complete

manuscript and there are missing pages, these will be noted. Also, if

unauthorized copyright material had to be removed, a note will indicate

the deletion.

Oversize materials (e.g., maps, drawings, charts) are reproduced by

sectioning the original, banning at the upper left-hand comer and

continuing fi'om left to right in equal sections with small overlaps. Each

original is also photognq)hed in one exposure and is included in reduced

form at the back of the book.

Photographs included in the original manuscript have been reproduced

xerographically in this copy. Higher quality 6" x 9" black and white

photogr^hic prints are available for any photographs or illustrations

appearing in this copy for an additional charge. Contact UMI directly to

order.

UMI A Bdl & Howell Infimnation Conqai

300 Noith Zeeb Road, Ann Aibor MI 48106>1346 USA 313/761-4700 800/521-0600

AB fivmo QUANTUM MOLECULAR DYNAMICS METHOD BASED ON THE

RESTRICTED PATH INTEGRAL: APPLICATION TO ELECTRON PLASMA AND

AN ALKALI METAL

by

Ki-Dong Oh

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF PHYSICS

In Partial Fulfillment of the Requirements

For the Degree of

Doctor of Philosophy

In the Graduate College

THE UNIVERSITY OF ARIZONA

19 9 9

DMI Niunber: 9927497

UMI Microform 9911 ASH Copyright 1999, by UMI Company. All rights reserved.

This microform edition is protected against unauthorized copying under Title 17, United States Code.

UMI 300 North Zeeb Road Ann Arbor, MI 48103

2

THE UNIVERSITY OF ARIZONA ® GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have

read the dissertation prepared by KI-DONG OH

entitled AR TNITTO miANTtlM MOLECULAR DYNAMICS METHOD BASED ON THE

RF.STRTCTKD PATH TNTKfiRAL: APPLICATION TO ELECTRON PLASMA

AND AN ALKALI METAL.

and recommend that it be accepted as fulfilling the dissertation

requirement for the Degree of Doctor of Philosophy

Robert H.,Parmenter Date nprember 23. 1998

December 23» 1998 loval W. Stark Date

December 23» 1998 Robert H. Chambers Date

^ C0)M^9J^iv\ i-L f i - j / Laurence Mclntyre Date

December 23» 1998 e A. Deymler Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

- DPfPinher 23. 1998 Dlssertatl(5n Director Pierre A. Deymler Date

December 23, 1998 -Dissertation Director Robert H. Chambers Date

3

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED

4

ACKNOWLEDGEMENTS

It is a great pleasure to thank to my advisor Prof. Pierre Deymier for his

considerate guidance and help from the beginning to the end of this work. Without his

encouragement amd patience, this work whould not have been possible. He shared his

expert knowledge and experiences, and made vital suggestions whenever I needed advice

and encouragement, amd I am extremely indebted to him.

A special thanks to my friend Frank Cheme with whom I shared the ssune lab,

same frustrations, and same happiness. The computer facilities in the Telecommunica­

tion Center at the Cornell University and the CCIT, Physics Department, and Material

Science and Enginieering Department at the University of Arizona were essential for this

work. I am greatfiil to Dr. Jadme Combariza for granting me generous computing time

and helpful tips of MPI programs.

Finally, I thauak my family, especially my wife Kim Soojung who gave me un­

conditional support and care over the years. Most of all I dedicate this dissertation to

my father and mother who are praying for us everyday.

5

TABLE OF CONTENTS

LIST OF TABLES 8

LIST OF FIGURES 9

ABSTRACT 13

1 INTRODUCTION 15

2 THEORY 18

2.1 Feynman Path Integral 18

2.1.1 Partition function for a single particle 18

2.1.2 Systems of interacting particles obeying Maxwell-Boltzmann statis­

tics 23

2.1.3 Two-Electron system 25

2.1.4 Many-Electron system 29

2.2 Path Integral with Non-local Exchange Using the Mean Field Approximation 33

2.3 Restricted Path Integral Method 35

2.4 Classical Isomorphism for Many-body fermionic system 38

2.5 Molecular Dynamics 41

6

2.5.1 Background 41

2.5.2 Restricted Path Integral Molecular Dynamics Method 46

2.6 Physical Quantities 48

2.6.1 Average of Physical Quantities 48

2.6.2 Energy Estimator 48

2.6.3 Correlation Functions 50

3 PRACTICAL IMPLEMENTATION 52

3.1 Treatment of Classical Potential Energy 52

3.1.1 Electron Plasma System 52

3.1.2 Alkali Metal System 55

3.2 .Algorithm of Molecular Dynamics 57

3.2.1 Overview 57

3.2.2 Creation of the Initial Configurations 60

3.3 Periodic Boundary Conditions in Path Integral MD 62

3.4 Calculation of Quantum Effects 66

3.4.1 Evaluation of det(£'^''''^) of the Effective Exchange potential with

PBC 66

3.4.2 Effective Force Calculation 70

3.5 Parallel Computation 71

4 APPLICATIONS AND RESULTS 76

I

4.1 Electron Plasma 76

4.1.1 Model System 76

4.1.2 Results 79

4.2 Solid/Liquid Transition of Alkali Metal 87

4.2.1 Model System 87

4.2.2 Results and Discussion 89

5 CONCLUSIONS AND FUTURE WORK 109

6 APPENDICES 111

6.1 .A.PPENDIX A : FREE R\RTICLE PROPAGATOR Ill

6.2 .\PPENDIX B : EXCHANGE KINETIC ENERGY ESTIMATOR ... 112

6.3 APPENDIX C : EXCHANGE FORCE CALCULATION 113

REFERENCES 115

8

LIST OF TABLES

4.1 Kinetic (A) and potential (B) energies per electron for various electron

densities. Kinetic energies per electron of current model are the average

values. Potential energies per electron are linearly fitted by using PE = a

-I- bT 83

4.2 Specific heat of the solid and the liquid, and latent heat. The experimental

values are at the data book(CRC). (1) and (2) in (A) are measure at T =

lOOK and 298K 90

9

LIST OF FIGURES

2.1 Necklace representation of a single particle for P = A. The number labels

indicate the different imaginary times 21

2.2 Exchange process between two electrons 27

2.3 Three-cycle exchange process among three identicle particles. Solid lines

and dashed lines represent two different exchange processes 31

2.4 Local(a) and non-local(b) exchange processes between two electrons. The

number labels indicate the imaginary times 35

2.5 Phase diagram of the sign of determinant. The diagonal line is x • y =

0. All unfilled dots correspond to positive density matrixes and the filled

dot has a density matrix with a negative value 39

2.6 Basic simulation cell(thick solid line) and its 8 image cells in two dimen­

sional system with PBS. The size of the basic cell is L. The radius of

the dotted circle is L/2. The dotted circle area is the effective range of

interaction 45

3.7 Coulomb interaction between two electrons i and j at different imainary

times 53

3.8 Empty core pseudopotential model for the ion-electron interaction. The

sum of the potentials (b), (c), and (d) is equal to (a) 57

10

3.9 Initial necklace configurations for two electrons in an electron plasma at

T = I300K and r, = .5. Only 1/3 of the beaxls {P = 480) denoted by filled

circles are shown in xy-plain 62

3.10 Beads of an electron necklace in the simulation cell and the image ceils.

Beads (1 and 4) are in the simulation cell and beads (2 and 3) which belong

to the same necklace are in the image cell. In the usual MD method, we

use the filled circles in (a) to calculate distances between beads. In (b),

the necklacp is reronstructed after translating beads (2 and 3) by L from

the left 63

3.11 PBC diagram with two electrons. The smaller filled dots represent the /th

beads of the electron j and the larger circles represent the berds of of the

electron i. We assume that the beads denoted by the large filled circles

are in the simulation cell 65

3.12 Algorithm for the calculation of the determinant det(£'^''-'^) 69

3.13 Sequantial job order in a parallel computer 73

3.14 Scalability of the parallel caiculation. CPU time scale is normalized to 1

for the serial calculation with one processor. .\ll time is measured at the

IBM SP2 in the Telecommunication Center at Cornell University 74

4.15 V'®"'' and potential energy in reduced units as functions of time steps. In

both cases, the thick lines and the thin dotted lines refer to skip = 10 and

skip = 0, respectively. 78

4.16 Kinetic energy of electron plasmas as function of number of beads in the

necklace representation of quantum particles. The circles and squares

refer to the high density (r, = 5, T=1800K) and medium density (r, =

7.5, r=700K) 80

11

4.17 Potential energy as functions of number of beads. The circles and squares

refer to the high density (r, = 5, r=1800K) and medium density (r, =

7.5, r=700K) 81

4.18 Kinetic energy as functions of temperature. The electron plasma with

ra=o, ra=7.5 and r5=10 are refered to by circles, squares and triangles,

respectively. The horizontal thick dashed lines correspond to the energies

of Ceperley[12,13]The thin dotted lines indicate the Hartree-Fock energies. 84

4.19 Potential energy as functions of temperature. The electron plasma with

ra=5, r,=7.5 and rj=10 are refered to by circles, squares and triangles,

respectively. The horizontal thick dashed lines correspond to the energies

of Ceperley[12,13]. The thin dotted lines indicate the Hartree-Fock en­

ergies. For both types of lines, r^^o, ra=7.5 and rj=10 are represented

from the top to the bottom 85

4.20 Iso-spin and hetero-spin electron-electron pair distributions for the high

density (r, = 5) electron plasma at T=1300 K (solid lines), T=1800 K

(dotted lines) and T=2300 K (dashed lines) 86

4.21 Running averages of electron kinetic energies at T = 273K and T = lOK.

The standard deviations are 0.003 (eV/electron) and 0.005 (eV/electron)

at T = 273K and at T = lOK, respectively. 97

4.22 Convergence of electron kinetic energy with respect to the number of beads

(P) in potassium. r,o„ = 273 K and Teu = 1300 K. Nion = ^eie = 54. . 98

4.23 Total energy of the potassium model versus temperature. The lines are

fits to the data in the low and high temperature regions 99

4.24 Various contributions to the total energy of the potassium system as func­

tions of temperature 100

12

4.25 Ion pair distribution functions at the different temperatures of 10K(thick

solid line), 76K(thick dotted line), 150K(dashed line), 248K(thin solid

line), 273K(thin dotted line), and 298K(thick long dashed line) 101

4.26 Trajectories of the potassium ions at (a) T=10K, (b) T=76K, (c) T=248K.

and (d) T=298K 102

4.27 Mean square displacement(MSD) of potassium ions as a function of time

and temperature 103

4.28 Vibrational amplitude of the potassium ions as function of the ion tem­

perature. The dotted line is a linear fit to the data except for the last

point 104

4.29 Normalized velocity autocorrelation function(NVAF) and associated power

spectrum for crystalline potassium(T=10K) and liquid metal(T=273K

and T=298K). The insert in the T=10K power spectrum is the exper­

imentally deduced phonon density of state at T=9K of a reference [19]. - 105

4.30 A 2D projection of the electron necklace (open circle) with potassium ions

(larsje filled circle) at T = 298K and T = lOK. The frame represents the

simulation cell 106

4.31 Partial electron-electron pair correlation functions. The solid lines and

dotted lines refer to the crystal at T=10K and the liquid at T=273K,

respectively 107

4.32 Ion-electron pair distribution functions at several temperatures 108

13

ABSTRACT

We develop a new Quantum Molecular Dynamics simulation method. The method is

based on the discretized path integral representaion of quantum mechanics. In this rep­

resentation, a quantum particle is isomorphic to a closed polymer chain. The problem of

the indistinguishability between quantum particles is tackled with a non-local exchange

potential. When the exact density matrix of the quantum particles is used, the exchange

potential is exact. However we use a high temperature approximation to the density

matrix and the exchange potential is only approximate. This new quantum molecular

dynamics method allows the simulation of collections of quantum particles at finite tem­

perature. Our algorithm can be made to scaJe linearly with the number of quantum

states on which the density matrix is projected. Therefore, it can be optimized to run

efficiently on parallel computers.

We apply this method to the simulation of the electron plasma in 3-dimensions

with different densities (r, = 5.0, 7.5, and lO.O) at various temperatures. Under these

conditions, the electron plasma are at the border of the degenerate and the semi-

degenerate regimes. The kinetic and potential energies are calculated and compared

with results for similar systems simulated with a variational Monte Carlo method. Both

results show good agreements with each other at aJl the densities studied.

The quantum path integral molecular dynamics is also employed to study the

effect of temperature on the electronic and atomic structural properties of liquid and crys­

talline alkali metai, namely potassium. In these simulations, ions and valence electrons

are treated as classical and quantum particles, respectively. The simple metal undergoes

a phase transformation upon heating. Calculated dynamic properties indicate that the

14

atomic motion changes from a vibrational to a diffusive character identifying the trans­

formation as melting. Calculated structural properties further confirm the nature of the

transformation. Ionic vibrations in the crystal state and the loss of long range order

during melting modify the electronic structure and in particular localize the electrons

inside and at the border of the ion core.

15

CHAPTER 1

rNTRODUCTION

Modeling and simulation have become a vital part of materials research. Modeling and

simulation techniques are maturing to the point where they offer hope for a practical and

reliable approeich for the study of real materials. The development of materials models

has evolved from the infancy of specific empirical descriptions, to highly accurate and

sophisticated representations based on first principle calculations. In the field of Ab-

initio molecular dynamics method, the method of Car and Parrinello[ll, 74], based

on the Density Functional Theory(DFT) has enjoyed a great popularity over recent

years. DFT molecular dynamics has been employed to investigate a very large number

of problems from condensed matter to chemistry to biology [65]. In constrast, applications

of molecular dynamics simulations using the discretized path-integral[23] representation

of quantum particles have been limited mostly to the simulation of systems containing

a small number of quantum degree of freedom (such as in the solvation of a single

quantum particle in a classical fluid[64]) or to problems where quantum exchange is

not dominant[50]. We should aiso mention the path-integral based method of Alavi and

Frenkel that allows for the calculation of the grand canonical partition function of fermion

systems[l]. With this method the fermion sign problem in the evaluation of the partition

function is solved exactly in the case of non-interacting fermions. When combined with

DFT, this method provides a means of doing ab initio molecular dynamics of systems

with interacting high temperature electrons[2].

Progresses in the simulation of fermionic systems by path-integral Monte Carlo

16

method[34, 35, 36, 15, 17, 95] have opened the way toward the implementation of a path-

integral based finite temperature ab initio molecular dynamics method(PIMD). In this

study, we describe such a molecular dynamics method applicable to the simulation of

many-fermion systems at finite temperatures. The method is based on (a) the discretized

path integral representation of quantum particles as closed polymeric chains of classical

particles (or beads) coupled through harmonic springs[23], (b) the treatment of quantum

exchange as crosslinking of the chains[18], (c) the non-locality of crosslinking (exchange)

along the imaginary time chains[34,35, 36], and (d) the restricted path integral[14, 15. 16]

to resolve the problem of negative weights to the partition function resulting from the

crosslinidng of even numbers of quantum particles.

The present PFMD is initially applied to the description of one-component

plasma, which consists of the electron gas with a uniform neutralizing background, at

the border of the degenerate and semi-degenerate regimes where the ratio of the tem­

perature to the Fermi temperature(7V) « 0.1. The electron plasma is the first focus of

our investigation because it is the simplest many-body fermionic system. It has been

extensively studied via path-integral, variational, and diffusion Monte Carlo methods

since the calculation of the equation of states of a Fermi one-component plasma such

as the interacting electron gas is a problem of fundamental practical importance as one

uses its properties in the density functional theory. The one-component plasma is also

a good prototype system as there exists a large amount of theoretical and numerical

data on its equation of state. The zero-temperature perterbative expansion of the en­

ergy of a three-dimensional uniform electron plasma in the high density limit, where

''j 1 (r, = r/oQ where r is the electron sphere radius and qq is the Bohr radius), was

calculated theoretically quite some time ago[28]. Accurate variational Monte Carlo cal­

culations have extended the zero-temperature equation of states of the degenerate Fermi

one-component plasma to a wide range of lower densities from r, = 1 to 500[12, 13]. The

exchange-correlation free energy has been subsequently calculated to encompass the full

range of thermal degeneracy[20, 66, 79].

17

After showing that the PEMD contains the necessary ingredients to simulate elec­

tron plasma up to metal densities at finite temperatures, we apply the PIMD to a simple

alkali metal, namely potassium K. We chose potassium because (1) it is a prototype

free-electron metal, (2) there exist experimental data for the pair correlation function of

a liquid potassium[89] and the power spectrum of crystalline potassium at 9A'[19], (3)

DFT molecular dynamics has had problems with metals when electrons leave the Born-

Oppenheimer surface and therefore violate one of the basic assumption of the method.

This problem has been solved technically in an ad-hoc manner with the introduction of

appropriate thermostats for the electronic and ionic degrees of freedom[8]. In the simple

metal case, the discretized restricted path integral representation of electron is the same

as that of the electron plasma. Classical ionic degrees of freedom representing potassium

ions are added to the model. We firstly show that the PIMD successfully models the

body centered crystal structure of the solid state of potassium at low temperature. Upon

increasing the temperature, the solid state transforms to a liquid. The predicted melting

temperature is below the experimental value. This deviation is assigned to a short-range

approximate form used in place of the usual long-range Coulomb potential in order to re­

duce computationcil time. The phase transformation is characterized thermodynamically

via energies, structually via pair distribution functions as well as dynamically via the

mean square displacements and the vibrational power spectra. Vibrations in the crystal

appear to induce some localization in the electron density. .A.s melting takes place, the

electronic structure responds to the loss of long range order in the atomic structure by

additionaJ localization.

We introduce the development of the path-integral molecular dynamics from a

one-body system to a non-localized many-fermion system in Chapter 2. In Chapter 3, we

explain the method of PIMD in further details and describe its practical implementation.

In Chapter 4, we present the results of PIMD calculation of electron plasma and an alkali

metal. Finally, in Chapter 5, we draw conclusions concerning the applicability of the

PIMD method to some other materials systems and suggest some future work.

18

CHAPTER 2

THEORY

2.1 Feynman Path Integral

2.1.1 Peirtition function for a single particle

Since Feynman[22] introduced the path integral of a quantum system, it has been well

developed[24, 25, 67] and applied to many-body systems[15, 18, 39, 58, 59, 60, 64]. The

basic idea of the Feynman path integral is to break a finite time interval into infinitesimal

time steps and then evaluate the matrix element of the propagation operator for each

step. In quantum statistical mechanics, all static properties and dynamic properties

of a system in thermal equilibrium are specified from the thermal density matrix. If

we work in the canonical ensemble, which is a system of fixed number of particles in a

fixed volume in equilibrium with a thermal reservoir, the probability of observing a state

with energy E is proportional to where ks is Boltzmann's constant and T is

the temperature. Let us consider a single particle system governed by the Hamiltonian

operator H. The partition function of this system may be written

Z = Tre ' ^"

= J dr <r|e~''^|r> (2.1)

where /3 is l / k sT and |r> is the exact eigenstate of H. In the path integral formalism for

many-body systems, we normally represent the matrix element in the partition function

as a density matrix, p{r{,rj;/3). The density matrix is defined as

p(r,-,rj;/3) = <r,|e- |rj> . (2.2)

19

Then the partition function can be rewritten as the trace of the density matrix

Z = J drp{r , r ; l3 ) . (2.3)

Before proceeding with the development of approximate forms for the density

matrix, we will first consider the matrix element in real time, f, for physicai clarity. By

substituting i{tf — ti)/h for 13, the density matrix p{ri,rf;0) for a particle governed by

the Hamiltonian, H, becomes

K(r j , t f ; r i , t i ) = <r (2.4)

where and t j are an initial time and a final time, respectively. The matrix element in

real time, K(r j,tf-ri,ti), which is the so called Kernel [23], is a solution of a real time

dependent Schrodinger Equation.

wherein the Hamiltonian H/ operates on the variables r/ and tj only. By analogy, the

density matrix in imaginary time is a solution of an equation of the form

dp -T3 = '2.6)

Eq.( 2.6) is a diffusion like equation. This fact will become important when we introduce

the restricted path integral and in particular, when we consider boundary conditions on

the density matrix. The kernel K{rj,tf\ri,ti) obeys the superposition principle, since

it is an exact solution of the Schrodinger equation in real time. By the superposition

principle, we mean that

= j drK{r j , t f , r , t )K{r , t ; r i , t i ) (2.7)

at any time t , where t i < t < t f . Eq. ( 2.7) indicates that one may calculate the matrix

element to any desired degree of accuracy for infinitesimal time interval, although the

matrix element, or the kernel A'(r/,i/;r,-,f,), cannot be calculated exactly for a finite

time interval, tj — ti. This is the basic idea of the Feynman path integral. In other words,

20

we calculate a matrix element only for each infinitesimal time interval after breaking a

finite time into infinitesimal intervals. The value for the finite time interval can be obtain

from the results of the evaluations for ail infinitesimal time intervals. Similarly, we may

evaluate the thermal density matrix p(r{, rj;/3) with appropriate accuracy if we divide

a finite temperature term 0 in P infinitesimal intervals, where P —¥ oo.

To evaluate the thermal density matrix, we will consider a set of P

different configurationa l s t a t e s , { | r , - > ; i = 1 , P} , where each s t a t e i s an e igens ta t e o f H.

With the relation

or

^ -0H ^

where £ —QjP, the partition function becomes

Z = J dr <rle-'^e-'^ • • • e-'"\r> .

Using the closure relation of the eigenstates |r,> of H,

J dvi Ir.xril = 1

and projecting the particle on (P— 1) intermediate states, the partition function can be

written as

^ = Jdridr2...drp <ri|e"''^|r2><r2|e~''^|r3> • • • <rp|e"'^|rp+i>, (2.8)

or

Z = j dr idr2 . . . d rpp{r i , r2 ;e )p{r2 , r3 ;e ) • •p{rp , rp^ . i ;€ ) (2.9)

where ri = rp^i indicates complete closure. Figure (2.1) illustrates a complete necklace

of a single particle for P = 4.

Each matrix element or each density matrix in the above relation represents

a propagator from one state to another state for infinitesimal imaginary time or very

21

«(.P)

Figure 2.1: Necklace representation of a single particle for P = 4. The number labels indicate the different imaginary times.

small deviation of temperature. In other words, the density matrix of a single particle

is connected to look like a polymeric necklace consisting of P beads. To evaluate the

density matrix of a single particle for infinitesimal imaginary time, we assume PT is

very large. Then we can adopt Trotter's second order appoximation[3]. According to

the Trotter formula[3-5],we have

e-"" = = Urn (2.10) P-+-00

and

g-e ( t+V ' ) smal l € (2.11)

where the Hamiltonian f l = T ->rV. T and V are a kinetic energy operator and a potential

energy operator , respectively. With Eqn.( 2.11), we can approximate the exact density

matrix by the product of the density matrix for T and the density matrix for V. The

error of this appoximation is of order of

Now we are going to evaluate the density matrix for infinitesimal imag­

inary time step c = l3/P. Let us assume that the Hamitonian H = T + K,where

T = —p^ jlm and V = K(f) is a local potential energy operator. Introducing a complete

set of momentum states, |p„> and using Eqn. ( 2.11), the density matrix becomes

A>(rn,'*„+i;6) = <r„|e~'=^|r„+i>

22

= J dPn<rn\Pn><PnW '^\rn+l> = yrfp„<r„|p„><pJe-^'/2-e-'^|r„+i>+C7(ee). (2.12)

After performing the momentun operator to |p„> and the local potential operator to

lrn+i>, then we obtain

/>(r„,rn+i;e) « (2.13)

where we define Po(»'n, ^n+i; c) as the density matrix of a free particle (also called the

free particle propagator), as

Po(r„,r„+i;e) = <r„|e"'=^|r„+i> (2.14)

or

Po(rn,r„+i;€) = J dp„ <rnlP„><Pnkn+I> e ^2.15)

Using the Gaussian integral,the free particle propagator becomes

PoK,r„«;^) = (^) ' exp . (2.16)

We give more details of the derivation of Eq. ( 2.16) in the Appendix A. From Eq. (2.9)

with £ = /?/P, the partition function of a single particle can be written as

r P , g Z = Yl 'drnp{r ,„r„+i ;—)

n=l ^

~ (2.17) n=l ^

or

with

(2.18) 11=1

P

V^jj(ri,r2,.:,rp) = XI* - r„+i)^ + pK(r„+i) (2.19)

The (*) on the product and the summation indicate that rp+i = rj. The partition

function of Eq. ( 2.18) is similar to a classical partition function. The first term of

23

the effective potential,Vg//, originates from the kinetic energy of a particle. We may

interprete this term as a harmonic type interaction between the nearest neighbor beads

in the necklace. The coupling constant is Ci = Pmlh^lS^. In the high temperature limit,

the necklace of P beads collapses to a single point so that a quantum particle becomes a

classical particle. The classical partition function is valid in the limit e = ^/P —»• 0. The

classical isomorphism is therefore more accurate at high temperature, T, and for a large

number of P. At low temperature, the quantum particle possesses some spatial extent

associated with its deBroglie wavelength.

2.1.2 Systems of interacting particles obeying MaxweU-Boltzmann statistics

In the previous section, we discussed the thermal density matrix and the partition func­

tion of a single particle in a canonical ensemble. Here, we will generalize them to systems

containing many quantum particles. We will attach particular attention to the contrast

between the discretized path integral form of the partition function of particles obeying

Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics.

The partition function for a iV-body system may be written as

Z = J dri - dr^fp{ri , - - ,r! \ !; l3) , (2.20)

where the thermal density matrix of N distinguishable particles, that are obeying Maxwell-

Boltzmann statistics, is defined by

p(ri, • • •,riv; /3) =<ri • • • ta^I \ri • • (2.21)

with |ri • • • ri\f> being an eigenstate of H. We will assume that the Hamiltonian of the

iY-particle system takes the form

iV -2 iV . jV » = E£ + E'>M + 5E«« (2-22)

«=1 t=l t>j

with Vij = v{ri — Vj) . 0(r,) is an external potential on the z-th particle and t>,j is a pair

potential between particles i and j. In order to evaluate the partition function, let us

24

discretize the density matrix by inserting (P — 1) intermediate states for each particle.

With the property of completeness of the eigenstates |ri of H. the density matrix

becomes

<ri • - -r/^l |ri • • • r^v> = <r\ • • • r;v| • • -e '^^ |ri • • -r,v>

= f n > . (2.23) •' j/=l n=l ifc=l

In Eq. ( 2.23), the subscript and superscript of denote the fcth element (or called

bead) of the necklace of the I'th electron. For the sake of convenience, we ^viH use a

new notation,]i2>= |ri and >= • •-rly' >. Using the Trotter

approximation and performing the potential operator, we may write the infinitesimal

density matrix as

= S-E." . > exp |-< '<•('•!") + E "("-I" - >•!") j I (2-24)

The term </?W| |fl(*+i) > in Eq.( 2.24) is a free particle propagator of N

distinguishable particles. If we use the result of the free particle propagator for a single

particle, Eq. ( 2.16), we have

rf) ><pS*^ . - -PS;)! e-'E.=. & |/2(^+M iV

, lt:\ .^llc\, fJtl fjfc^ fit) (k\.

n=l =/n

n=l

= /n • • -pSv" ><p1" • •

If we repeat the evaluation of an infinitesimal density matrix over all intermediate states,

the partition function of the A'^-body system can be written as

/r' \ 3NP/2 . P iV ^« (t) / n n (2.26)

1=1 i^i

25

where ^ P N

(2.27) ifc=i t=i ifc=i t=i

and

E ^ E i; "(••!" - r™). * 1 I .• 1 * f__. -<r=l i=l fc=l i>j

(2.28)

The effective potential, Vi + V2, of the N-body system in absence of quantum exchange

is similar to the effective potential of one-quantum particle system. Vi represents the

harmonic potential which corresponds to interactions between the first neighbors in the

closed necklaces. V2 is nothing but the potential energy resulting from the exLernal field

and the particle/particle interactions.

2.1.3 Two-Electron system

In the previous section, we have established an isomorphism between a classical par­

tition function and a path integral of iV distinguishble particles. To extend this to

many-fermion (and many-boson) systems, we first investigate a system of two electrons.

Subsequently, we will generalize the system to iV indistinguishable fermions (or bosons).

Since identical particles can not occupy the same state by the Pauli exclusion

principle, the total wave function of a two-electron system should be antisymmetric upon

exchange between electrons. Using this fact, we will introduce a new space in order to

represent a state of two indistinguishable fermions. The new space is defined as

with one particle in state ri and one particle in state r2. The closure relation of this

space is

(|rir2> -|r2ri>. (2.29)

^ Jdridr2 |rir2}{rir2| (2.30)

The density matrix of 2-electron system can be written as

p(ri ,r2; r'i,r^;/3) = {ri,r2|e (2.31)

26

If we consider an intermediate state jr'jrj} with the closure relation, we have a convo­

lution relation for two identical particles such that

P(ri,r2; = J dr"dr2 p{ri ,r2 ; r",r2; (3/2) p{r ' ( ,r2; r ' i ,r2; 012). (2.32)

We recall that the partition function is the trace of the density matrix and

Z = Jdridr2p{ri ,r2; ri ,r2;f3) (2.33)

Using the convolution relation,Eq.( 2.32), for (P — 1) intermediate states in imaginary

time, the partition function now becomes

Z = /n n ('1". 4" : "; <) (2-34) •' l/=l fc=l

where r,- = with i =1, 2, that is, each electron forms a closed necklace

with P nodes. Let us evaluate an infinitesimal density matrix with Eq.( 2.29).

= 5 (<rfV<"|e-'»|rS'+'Vf+"> + -

To evaluate the last terms of Eq.( 2.35) including the cross terms between particles, we

will consider a general case.

=<rl'lr['l| (2.36)

and the term in Eq( 2.36) is a density matrix of

a free particle propagators, which can be evaluate exactly in the same way as for the

classical particles obeying Maxwell-Boltzman statistics. With the results of Appendix

A, we have

(2.37)

27

Eq.( 2.37) shows that

<rWr[""| (2.38)

If we consider a symmetric potential^such as a pair-wise additive central potential.

Eq.( 2.35) can be simplified as

(2.39)

From Eq.( 2.34- 2.39), the partition function of two electrons finally becomes

\2NPI2 c P , r ^ = (^) / n 11^ ^ ' ' ' ' ^

^ ^ ^ „=i fc=i I

—i3-L o pV^ (2.40)

The exchange process between the two electrons is shown in the figure (2.2). By factoring

o— k k

t-

Elactrani k^l ElictronZ

Figure 2.2: Exchange process between two electrons.

the first term out of { } in Eq.( 2.40), we have

X 11 "**2*^'' )^) + )2 j

28

For convenience, we may rewrite this term by considering the product terms as

Ilf } = >+!'•'-'•= 'i . fc=l k=l k=l

= exp {-aCo t, E (>•!" - • n let (e<'-'+'I) (2.41) \ 1=1 k=l / fc=l

with

-0Co ( 1 e ^ '

-3Co ((r^*' -rj*"*"'' )2+(r^*' ; \ / 1

det (£(<^-*=+1)) =

From Eq. ( 2.40 and 2.41), the partition function of two-electron system becomes

^ 1/=! \ t=l Jt=l /

X n <!« (£<*•'+'1)6-'^=""' (2.42) Jc=l

We note that an infinitesimal density matrix of two electrons between the imagi­

nary times k and k-i-l is propotional to det so that the sign of the infinitesimal

density matrix can be either a positive number or a negative number. This is the origin

of the sign problem in the simulation of many-fermion system. In particular, the parti­

tion function (2.42) is an integration of terms which can be either positive or negative so

one can think of the partition function as the difference between a large positive num­

ber and a large negative number to give an overall positive value. Therefore the noise

level of the partition function is large and this is the difficulty to overcome during the

simulation of a many-fermion system. Furthermore since det can be negative,

it is not possible to rewrite Z in the form of a classical partition function. Thus one

cannot establish yet an isomorphism between the fermion system and a classical one. In

order to solve the sign problem, we will adopt the fixed-node path integral method in

our calculation. The details of the fixed-node path integral method and its usage in our

simulations of many-electron system will be discussed in section 2.3.

29

2.1.4 Many-Electron system

We may extend the partition function of the two-electron system, equation (2.42), to

the partition function of a indistinguishable /V-body quantum system. Because of the

indistinguishability between particles, we will introduce an orthonormaJ basis of the

indistinguishable iV-body fermionic system:

|rir2---riv} = J^(-l)«'|rpirp2 • • •rpiv> (2.43)

where \r1r2 is an orthonomal basis of an distinguishable iV-body system, and p

is the parity of the permutation and (-1)" become +1 and -1 for even and odd number

of the permutations between fermions, respectively. The closure relation of the new basis

becomes

1 ^ • • •r^}{rir2 • • - Vtv] = 1- (2-44)

1=1

With the new basis, we may define the density matrix of the indistinguishable iV-body

fermionic system as

p{R,Ii';3) = {R\e^"\R'}, (2.45)

or

p{R,R';0) = (2.46) p

where |/Z} = |rir2---riv} and |/2> = |rir2 • • - r .v >• If we consider P intermediate

states and the closure relation, (2.44), we may have the following convolution relation

for the density matrix:

p{R,R';0)= f (2.47) . = ! where = p{R^^\ e), R^°^ = R, R^^^ = R', and e = I3IP, which is the infinites­

imal imaginary time interval.

30

The partition function, which is the trace of the density matrix, for an indistin­

guishable iV-body quantum system may be written in the form

Z = J dRp{R, R: 13)

= [{[dR^^^ (2.48) t=0

In order to calculate the partition function, we have to evaluate the infinitesimal density

matrix, p(R^^K t). For further development, we will assume that the Hamiltonian

of the system is H = t + Vjt where again

.V -2

t=i

i=l i.J

(i>{ri) is an external potential at ri and y(|r, — rj|) is a pair potential between particles

i and j. With the Trotter's approximation, the infinitesimal density matrix becomes

P

P

If we apply the closure relation,/dR^''^R^''^><R^''^ = 1, and the orthonormal property

of the basis >, the infinitesimal density matrix can be written as

= po{R^''\R^''-^^^;e) (2.49)

where the infinitesimal density matrix of free particle is defined as

p

The infinitesimal density matrix of free particle contains the exchange procceses between

identical particles and it can be written in a determinant form. Figure 2.3 From the

equation (2.25), the infinitesimal density matrix of free particles becomes

(-C.B'-S' .

31

I [

ele 1

(k+l) 1

I I

j

\

1

ele 2 ele3

t (

I

i

i I

(k+l) (k+I)

Figure 2.3: Three-cycle exchange process among three identicle particles. Solid lines and dashed lines represent two different exchange processes.

32

with C2 = m/2eh^. If we perform the summation with the permutation operator p, we

can simplify the density matrix by using a determinant:

/ r \ NPii e) = det , (2.50)

where the matrix is defined as

Furthermore, the exchange processes between particles can be illustrated by writing the

infinitesimal density matrix of free particle (3-particle case) as

y ^ k 3 N P / 2 e) = (^2l^y ^0 • (1 — f i2 — f23 — hi +ff23i +^312)1 (2-51)

where

1 - C 2 f ( r " = ' - r ' ^ + " V - C 2 f ( r ' ' " - r ' ' = + " ) ' - c ^ V Qiji = —.e ' J € y ' ' ' e V ' • / . (2.52)

/./s and giji's are the exchange process between two particles (t,j) and the exchange

process among three particles respectively. We show the exchange between iden­

tical particles in fig. (2.3). The determinant of the density matrix of the free particles

in absence of quantum exchange is factored out of equation (2.50);

det = IJ • det ^ (2.53) t=i "

where all the exchange effects (including the sign of the density matrix) are included

in det(£'(*^'*+^') which elements are defined as (£'^^'^"''^^),j = (A^^'^'^"''^'),j/(j4^^*'^"'"^'),',.

Specifically, the matrix element of det(£;'^^''^"^^) for AT-particle system is given by

=exp [-5C. {(r!'l - (rf (2.54)

33

From equation (2.48,2.50,2.53,2.54), we finally write the partition function of the N-

fermion system as

/ r- \ 3NP/2 r P / . V P , \ = (^) / n n ''-i" -p (-i co E E (-1" - -1' ")) ^ •' \ i=ik=i J

X JJ det (2.55) fc=i

where the classical potential energy Vi is the same with equation (2.28), i.e.

= 4 E Z «>(••!") + 4 E E "(--l" - --f)- (2-36) fc=l i=l k=l i>j

2.2 Path Integral with Non-local Exchange Using the Mean Field Approx­

imation

In the preceding sections, we have developed a local non-interacting density matrix which

does not describe electron correlation, since the free particle density matrix has been

obtained by using a complete set of states represented by Slater determinants of plane

waves and Slater determinant of plane waves are solutions to the Hartree-Fock equation

for free electrons. Although the local non-interacting density matrix does not include

electron correlation, in the limit of high temperature, its nodes approximate reasonably

well those of the exact density matrix[15].

We now construct an approximate form for the density matrix that includes

electron correlation. In order to treat the correlation between like-spin electrons, Hall

has proposed a non-local exchange pseudopotential[34, 35]. In the local form of the

density matrix (Eq. (2.51)), det(£'(*^'^+^)) includes ail the exchange effects. Although

the exchange occurs only between consecutive beaids in imaginary time. Furthermore, in

the limit of 6 —f 0, the matrix converges to the identity matrix and the system

collapses into a bosonic state. To avoid this undesirable behavior and inspired by the

consideration of quantum chemistry simulation, where it is well known that exchange is

a non-local interax:tion in space, Hall has suggested a non-local form of a density matrix

34

of two-electron system as follows;

P -»• n <let(£'i*'-')), (2.57)

1=1

where the matrix element (£'i*'''),j is definfed as

= exp (-^ Krfl - rl'V - (rl" - ri")^|). (2.38)

In the preceding relations, the superscripts and the subscripts label the beads and elec­

trons, respectively. We note that a in Eq. (2.57,and 2.58) is a system dependent pa­

rameter and the absolute value of the argument of the exponential prevents negative

weights. Because it is not easy to find a proper parameter a for a system and it is clear

that one underestimates the contribution from negative values by choosing the absolute

value, we generalize the non-local density matrix of an jiV-fermion system by choosing

the following;

det (.4(^-^+1)) n ,, • n^et . (2.59) i=l " 1=1

The non-local form of the infinitesimal density matrix of an iV-fermion system and the

corresponding partition function now becomes

, r \3NPI2 S P ,p det (2.60)

and

/ r> \ zNPn . P N / N P , \

^ u=lj=l \ i=lJc=l /

x f [ f l d e t (2.61) /=l *r=l

The non-local form for the density matrix cannot be obtained from simple Slater de­

terminants of plane waves. Equation (2.60) should therefore represent electrons beyond

the Hartree-Fock approximation. A non-local density matrix would account for some

electron correlation. In Fig. 2.4, we illustrate both the local exchange process in (a) and

35

ele 1

ele 2

5 . . . P 1 4 3 2

(b) ele 1

ele 2

1 2 3 4 5 P

Figure 2.4: Local(a) and non-locai(b) exchange processes between two electrons. The number labels indicate the imaginary times.

the non-local exchange process in (b). In the non-local exchange model, a bead of an

electron interacts with all beads of any other electron.

Although the non-local form of the density matrix does not collapse to a bosonic

state when P —> oo, it still has the sign problem, because the determinant vaiues are

either positive and negative. In the next section, we will discuss and present a solution

to the fermionic sign problem by introducing the restricted path-integral Monte Carlo

method. This method has been widely used in the Monte Carlo simulation but never

used in the molecular dynamics.

2.3 Restricted Path Integral Method

There is a fundamentai difficulty in the simulation of many-body fermionic system, called

the sign problem. The sign problem arises from permutations between identical particles.

36

The contribution from even permutaions is almost the same as the contribution from odd

permutations. In practical calculation of thermodynamic properties, one can not expect

accurate results, because of large signal to noise level. The study of the sign problem

and the search for better conditioned simulation algorithm are widely discussed subjects

in the simulation of many-electron systems [7, 2, 14, 15, 53. 90].

In more recent studies, a restricted fixed-node path integral approximation has

been suggested to solve the sign problem on a many-fermion system with path integral

Monte Carlo simulation. In this approximation, the paths of all fermions in time( imag­

inary time) are restriced to remain within the region of phase space where the density

matrix is positive. This approximation has been applied to liquid ^He above 1A'[15]

and the hydrogen plasma at high temperature and reasonable agreement to the existing

theories has been found. On the basis of these two studies, we may be able to restrict

phase space of electrons to the region where only positive density matrices are allowed.

The main idea of the restricted fixed-node is originated by Metropolis and

Ulam[52, 87], who have suggested the extansion of the random-walk process typically

used to simulate the diffusion equation for solving Schrodinger equation. Anderson[7] has

applied the restricted fixed-node scheme to obtain the ground state of simple quantum

molecular systems. .Anderson also solves Schrodinger equation using the random-walk

methods. We will summarize the restricted fixed-node path integral idea suggested by

Ceperley[14, 15, 16] and then apply it to the simulation of fermion systems by path-

integral molecular dynamics.

where C = C{x,t) is a concentration, D is a diffusion coefficient and i is a real time.

The solution of the diffusion equation without any restriction is

if we assume that the initial condition is C{x,t) = S{x — iq). As we mentioned in

A diffusion equation in one dimension can be written as

(2.62)

y/AnDt (2.63)

37

the section 2.1, the diffusion equation is isomorphic to the imaginary time-dependent

Schrodinger equation of a free particle density matrix:

dp{x,^) h} aV(x,/3) d0 2m dx^

by replacing i3 by it/h.

(2.64)

If we assume that the boundary condition of the diflfusion equation,

C(x-z') = 0, (2.65)

then the solution of the equation ( 2.62) becomes

1 Ix—zn—x')^ 1 (x+xn—i')^

The solution ( 2.66) has an antisymmetric form in space about x' and we can

impose an infinite potential barrier at the boundary x, which will be called the fixed

node, without changing the solution because the solution of equation ( 2.62) is uniquely

determined by the boundary condition. By virture of the isomorphism between the dif­

fusion equation and the imaginary-time Schrodinger equation, we may apply the infinite

potential barrier at the node of the density matrix, with the fact that the trace of the

density matrix should be a positive real physical quantity and be spatiaJly antisymmetric

if exchange process occurs between like-spin electrons. In the path integral, the trace

of a density matrix p{x,x;(3) is always positive, but the infinitesimal density matrix

p{xi, x,+i; T = (3/P), where i = 1 to P, can be either positive or negative, we can choose

a configuration of beads of a necklace such that x, and x,+i are on the same side of the

boundary node so that p(i,-, x,+i; r) for all i are positive. In the fixed-node Path Integral

Monte Carlo method, one begins the simulation with a trial density matrix, whose nodes

are known. So if the trial density matrix is exact, the method then becomes exact.

In contrast to the Monte Carlo method, we start the simulation with arbitrary

configuration of beads of electrons without the knoweldge of the nodes. We choose

only the configurations whose determinants are positive so that all infinitesimal density

matrices have positive values. Then we calculate the forces ( called exchange forces) of

38

the configurations to generate a new configurations. More details will be given in the

next chapter.

To see the meaning of the fixed-node and to understand the sign problem more

clearly, we consider the special case of a two-body system in 3-dimensions with an ex­

change process between a pair of beads (i) and (j). An element of infinitesimal density

matrix is

where and are the coordinates of electron 1 and 2, respectively. Then the

determinant of the matrix is

12 = J g-Ci {(x<'' -y'-'' -x<J> )2}

g-Ci -y'-*' )^} i

With relative coordinates x = — y^'^) /y/2ci and y = — y^-'h/V2ci, the deter­

minant can be rewritten as

det(E('-'>) = 1 - e-^ y (2.67)

In the relative coordinates, the sign of the determinant now has the same sign with the

dot product (x • y). The boundary between the two regions is x • y = 0. The figure 2.5

shows these regions clearly.

2.4 Classical Isomorphism for Many-body fermionic system

In the numerical simulation of the Many-body fermionic system, we are not evaluating

the partition function directly. Instead we are trying to establish an isomorphism between

the quantum partition function and a classical partition function. With the restricted

path-integral method, we can always constrain the configurations of the particles in the

system to regions of phase space with positive density matrix. We therefore rewrite the

39

«+/

Figure 2.5: Phase diagram of the sign of determinant. The diagonal line is ® • y = 0. All unfilled dots correspond to positive density matrixes and the filled dot has a density matrix with a negative value.

40

partition function of the many-body fermionic system, Eq.(2.61), in a form isomorphic

to a classical one:

where the integration is limited to configurations with det(£'''^*'^) > 0. In equation (2.68).

V2 is a classical potential energy funtion of the position, which can be written as

V2 = V2(rj^^), (2.69)

where = |r-*' — denotes the distance between the Arth beads of the electrons i

and J. Since the ions in metal system will be treated in a classical manner, the distance

btween the ion / and he Arth beads of the electrons J is = |r/ —Ve// in equation

(2.68) is a quantum effective potential energy. The classical potential function includes

electron/electron Coulomb interactions or electron/ion interactions. In contrast, v;//

includes quantum exchange energy between electrons. From equation (2.68,2.61),we can

define the effective potential,Kg//, as

y.// = y.fr+K?/'- (2.70)

where xz-harm _ P ^ /•« -.1 ^ ) (2.11)

and

K."/' = £ E (d«(e"-'>)) , (2.72) ^ fe=l /=1 ^

where det(E^'^''') > 0 and P" is the effective number of paths with det(E^^''') > 0.

Y^arm jg non-exchange harmonic potential and is a non-local quantum exchange

potential. In the non-local form of path-integral dynamics, a quantum particle is still

represented by a necklace of P beads such that a point in the necklace interacts with

its next consecutive neighbor along the chained necklace through a harmornic potential

with a strength mP/20^h^. In contrast to the harmonic potential, the exchanges between

different particles are not limited to the nearest neighbors along the necklaces, but act

over all beads of the different necklaces.

41

In order to represent with a more practical form for implementation of

a restricted path-integral molecular dynamics, we introduce a step function d"*". The

function 9^i ensures the path restriction by taking on the values I and 0 for paths

with positive and negative det(£'^*'''), respectively. In addition, for a system containing

electrons with two diflFerent types of spins (i.e. spin-up and spin-down electrons), we may

use the fact that the density matrix is approximated as the product of two determinants

taking the form of equation (2.72); one determinant for the electrons with one type of

spin and another determinant for the electrons with the other type of spin[37]. We now

rewrite the as

1 down PP.

n?;" = -3EEE^I'' (detCEC'l)) «5,. (2.73) ^ «=«p it=l 1=1 '

where

p:=j:{:oL-k=i 1=1

In the previous two equations, s in the summation denotes the spin of the electrons.

Up to now, we have written the partition function of a quantum fermion system

in the form of a classical partition function. In the following section, we will use molecular

dynamics to sample the newly developed classical-like effective potential for a quantum

system.

2.5 Moleculeur Dynamics

2.5.1 Background

If one has a microscopically well-defined physical system, one can use Molecular Dy-

namics(MD) method to calculate the physical properties of the system. MD method

computes phase space trajectories of a system of many particles which individually obey

the classical Newtonian equations of motion. Specifically, if a iV-particle system is de-

scribed by a classical Hamitonian H = + Et>i where o,- is a velocity of a

42

particle i and (pinj) is a pairwise centeral potential between particles i and j separated

by Pij = |r,- — Tjl, one can anaijrtically specify the phase space trajectories, (r,(t),

by solving the Newtonian equations of motion with a certain initial conditions. In other

words, we can find the time evolution trajectories by solving the following equations;

. .9,

and ^ at

where Fi is the total force on the particle i, which is the sum of the forces on a particle

i from all the other paticles in the system, and also the force on the particle i from a

particle j, can be obtained from f^j = —In order to solve the equations

of motion in a numerical MD simulation, one discretizes the differential equations of

motion. Several numerical schemes are then used to integrate the equations of motion.

Among these schemes a finite difference method is often used.

Since Alder and Wainwright[4] used MD method for N-body system, the MD

method has been developed and applied to simulate a large variety systems[10-24]. The

main advantage of the MD method over the Monte Carlo method is that it allows the

calculation of time-dependent properties in addition to the equilibrium properties which

can be obtained by either methods.

Early simulations were carried out for systems where the energy was a constant of

motion[4, 5, 70, 86]. Accordingly, properties were calculated in the microcanonical en­

semble where the number of particles, the volume, and the energy were constants. The

other model of the MD method is the constant pressure method. This method was intro­

duced by Andersen[6] and subsequently generalized by Parrinello and Rahman[62, 63, 56].

The volume of the system is treated as an additional dynamical variable in this method.

The MD method of the constant pressure is applied to structural changes in the solid

state. However, in most situations one is interested in the behavior of system at constant

temperature. This is partly due to the fact that the appropriate ensemble for certain

quantities is not the microcanonical but the canonical ensemble. After W6odcock[93]

used a constant temperature MD with a momenturm rescaling procedure, in which the

43

velocities of the particles are scaled at each time step to maintain the total kinetic

energy at a constant value, several constant temperature MD methods have been pro­

posed. Haile et a/.[32] have examined analytically the constraint method based on the

momentum scaling procedure. Andersen[6, 48] proposed a hybrid of MD and Monte

Carlo method and introduced stochastic collisons in the phase space trajectories. .\n-

other constant temperature MD Method has been proposed by Hoover et a/.[41. 45]. In

this method, the force on a particle is constrainted such that the total kinetic energy is

constant. Introducing virtual variables, Nose[56, 57] has generalized the constant tem­

perature MD method.

In our MD simulations, we adopt the constant temperature MD method based on the

monentum rescaling procedure. In this method, the equilibrium distribution function

deviates from the canonical distribution by order of N~^ for a N particles system [57].

Thus the average quantities calculated with this method will be in error of C?(iV~'). In

the momenturm rescaling constant temperature method, the momenta of the particles

in a simulation cell are rescaled at each time step to maintain the total kinetic energy

at a constant value. If T/ is an instantaneous temperature of the system,

where KE is the total kinetic energy of the cell and p,- is the momentum of the Jth

particle. Using this relation, the momentum can be rescaled such that

In molecular dynamics simulations, we generally restrict our particles within a

physical volume, the basic cell or the MD cell, to retain a constant number density of

the particles of the system. Let the system consist of N particles within a cubic basic

cell which an edge length , L, and a volume ft = When the system evolvs in time,

KE

(2.74)

,scaled

where yJTrej/Ti is the scaling factor and Tref is the desired temperature of the system.

44

particles will hit the surfaces of the cell and be reflected back into the cell. Especially

for the system with a small number of particles, we would have some unexpected effects

from the restricting surfaces. In order to reduce the surface effect we impose periodic

boundary conditions(PBC)[80]. With PBC, the MD-cell (or the basic simulation cell) is

copied an infinite number of times by identical image cells. We may, therefore, write a

physical quantity A(r) in the MD-cell as

A(r) = A(r - \ - n • L ) (2.76)

where n = (wi, nj,,nj) whose components are integers, L = { L x , L y , L z ) is the size of

the simulation cell, and r is confined within the cell, i.e. [r| < L. Equation (2.76) means

that if a particle crosses a boundary of the cell, it re-enters through the opposite side of

the cell at the same instant. Figure 2.6 shows the behaviors of particles with the periodic

boundary conditions in two dimensions. There are 26 image cells in three dimensions.

With the periodic boundary conditions, the finite MD-cell is extended to infinite by

indentical image cells. In other words, due to the PBC the potential energy is represented

by

^ H L \ ) , (2.77) i < j n i < j

where $(r,j) is a pair potential, = |r,- — rj|, and r, and rj are restricted with in

the MD-cell. In order to avoid the infinite summation in the last term in Eq. (2.77),

we introduce a cut-off range (r^) for the potential[92, 40]. .A. particle in the basic ceil

interacts only with each of the N - I other particles in the MD-cell or its image cells.

Effectively we may cut off the potential at a range

Tc < L/2. (2.78)

With this cut-off range rc, we rewrite the potential energy as

^ = 1 2 E - • = ) ) ( 2 r a ) t (< j ) j^MDcell

In figure 2.6, we illustrate the cut-off range of the Arth particle by a dotted circle. We as­

sume that the jth and the fcth particles remain in the same positions but the ith particle

45

O k O k C k

C- ..

i O j i'

^O-

i o j

r-i o ••

i'

Ok ^k O k

0 . 1 O

J '' i

j i-

.O:

i O

j i'

o'' o O k

a

i ••

j i* i "O

j i-

JD- . . i • o

j i'

Figure 2.6: Basic simulation cell(thick solid line) and its 8 image cells in two dimensional system with PBS. The size of the basic cel l is L. The radius of the dotted circle is L/2. The dotted circle area is the effective range of interaction.

46

moves from a site i to another site i' after some time At. At time t, the Arth particle was

affected by the ith particle in its image cell, not the ith particle in the MD-cell. However,

potential between the ith and the Arth particles is calculated within the MD-cell at time

t + At.

Although we can effectively simulate a system with MD method with PBC and a trun­

cated potential, such a truncation may have a determinential effect on systems with the

long range interactions such as systems of particles interacting via the Coulomb poten-

tiai. There is a very elegant procedure, known as Ewald's method, to calculate effectively

Coulomb potentials with PBC[43, 71, 94]. The Ewald's methods will discuss in section

2.5.2 Restricted Path Integral Molecular Dynamics Method

In this section, we develop a classical Hamiltonian useful with molecular dynamics sim­

ulations based on the restricted path-integral representation of quantum particles. Once

we set up the Hamiltonian of a system, we can calculate the phase space trajectories of

all particles with the molecular dynamics method.

Let the system consist of Nei unpolarized electrons. For the system, the num­

ber of spin-up electrons is the same as the number of spin-down electrons. If there is

no external force, the total potential for the system is the sum of the electron-electron

Coulomb potential and the effective potentials given by equation (2.71,2.73). By con­

sidering a kinetic energy term for the N^i electrons, the classical Hamiltonian can be

written as

3.2.

"el r- r i ,2 i

1=1 k=l k=l i{>j) j=l |r, -, down P P 1

(2.80)

where P P

k=l 1=1

47

In Eq. (2.80), m" is an arbitrary niass of a bead (we chose m" = 1 a.m.u.)[64] used to

define an artificial kinetic energy for the quantum states in order to explore the effective

potential surface, Vejj, and rric is an electron mass. We recall that the subscript and

the superscript on the position are for labelling an electron and a bead in this electron,

respectively. The second term in the above equation accounts for the electron-electron

Coulomb potential energy. The forces derived from the non-local exchange potential,the

last term of Eq. (2.80), are calculated as means over the restricted paths with positive

determinants. Therefore, an effective force calculation requires a satisfactory sample

of such paths. Since the exchange potential offer a barrier to paths with negative de­

terminants, it biases the sampling of phase space toward configurations with positive

determinants. Although many configurations with negative determinants may exist and

evolve, they do not contribute to the exchange forces.

To extend the restricted PIMD method to cdkali metal system, we simply add

the potential terms related to the classical ionic degrees of freedom to the Hamiltonian

of the electron plasma system. For a system containing Nei unpolarized electrons and

^ion ions, the Hamiltonian becomes

^ton 1 ^lon ^ton I H = H.,.+ £

1>{J) J

^ton ^ton t + E E E p^pWo(«/ - r!'^^), (2.81)

/=l 1=1 /=l ^

where M/ is the ion mass and R is the position of an ion. In Eq. (2.81), Heu is the

Hamiltonian of the electron plasma described in Eq. (2.80) and the second term is the

kinetic energy of the ions. ^(R[j) and V^seurfo are the ion-ion potential and the ion-

electron pseudo potential, respectively. These two potential energies will be discussed in

section 3.1.

48

2.6 Physical Quantities

2.6.1 Average of Physical Quantities

Using the molecular dynamics method, one can obtain the phase spcice trajectories,

( r(i), p{t)), and subsequently any microscopic physical quantites, say A{r{t),p{t)), as

functions of time. If the system is ergodic, we may compute the time average of A{t) to

find its corresponding macroscopic property. In other words, the average of the physical

quantity A can be calculated as

2.6.2 Energy Estimator

We have constructed the canonical partition function for a many-electron system with

P intermediate states in the imaginary time. To evaluate the mean energy, let's rewrite

the partition function (Eq. (2.68) ~ (2.73)) for the convenience:

where the potential energy Vj includes all classical potentials such as the short range

interatomic potential and the pseudopotentiaJ between an electron and an ion as well as

the long range coulombic potential. The non-local quantum effective potential, is

given as

/ A { r { t ) , p { t ) ) d t . J o

(2.82)

where down P P

Now we can evaluate the energy estirmator by

(2.83)

49

The estimator for large enough P becomes

IE\ - /V / /V p

(EEt-: \ j= l Ar=l

)') + (^(/'KTf)) + <^2> • (2.84)

In Eq. (2.84), the angled brackets denote ensemble average and these quantities will

be evaluated by time average calculation, Eq. (2.82). Since the last term of the energy

estimator equation can be interpreted as a potential energy estimator, the kinetic energy

estimator, (KE), becomes

The first term in the left-hand side of Eq. (2.85) is a constant for given P, iV, and T. The

second term of the equation is the harmonic eiFective potential contribution to the the

kinetic energy estimator. Both terms are easily calculated with minor computing time

if all configurations are known. However, the third term in the equation, i.e. {KEexch}^

which is the exchange potential contribution, cannot be calculated unless we pay a major

computing cost, because the number of required operations is the order of ~ iV^P^. We

will give a full derivation (KEexch) term in ."VPPENDIX 2 and simply write the result

here:

(2.85)

where (KEexch) is defined by

j n down P P , \ = (^EEi :^Ndet (E( '=- ' ) )0L)

\ a=«P fc=l /= l ^ /

(2.86)

(KEexch.) — 2^ 2->

down P P (2.87)

3=up k=l 1=1

where is a.n N x N matrix and its element (^i*^''^)at is given by ik,l)

(2.88)

and

e " = ( rW-r ' " )=-( r (" - r ( ' )p .

50

2.6.3 Correlation Functions

Pair Correlation Function

An important physical quantity that we can easily compute during a simulation is the

pair correlation function (also called the radial distribution function), g{r), which tells

us the actual spatial distribution of one kind of particles with respect to another kind

of particles. With the pair correlation function, we can compare simulation results with

the experimental structural data, like that obtained from X-ray diffraction. The pair

correlation function between the (i) and {j) types of particles ,^,_j(r), is defined as

where Q is the volume of a simulation cell, N j is the number of the type j particles, and

n,(r) the number of the type i particles situated at a distance r and r + <5r from a type

j particle.

Time-Correlation Functions

We often calculate the time-correlation functions to understand macroscopic transport

properties of a system. One of the time-correlation functions is the mean square dis­

placement (MSD). The mean square displacement is defined as

rT M S D { T ) = lim / (r(«-I-r) - r( r ) ) ) 2 (2.90)

T->oo Jo

with r being a correlation time and r being the position of a particle. By calculating MSD

from the trajectories of particles, we may determine whether the processes developed by

a system is a diffusion process or a vibrationaJ one. For normai diffusion processes, i.e.

one that occurs slowly with respect to microscopic times and for which spatial variations

are smooth, we have

D = ^-MSD{t), (2.91)

51

where D is the coefficient of self-diffusion[51]. The mean square displacement is a linear

function of time for a liquid.

Another time-correlation function is the normalized velocity autocorrelation

function (NVAF), which is defined as

where the velocity autocorrelation function (VAF) is given by

VAF(r)= Um / (t»(t + r) - tj(r)))2. (2.93) T- oo Jo

V in Eq. (2.93) is the velocity of a particle. Using the velocity autocorrelation function,

we can also calculate the power spectrum or spectral density G{f) of the NVAF. By the

Wiener-Khintchine theorem[42], the power spectrum is the Fourier cosine transform of

NVAF, i.e.

G(/) = lim 4 [ NVAF(r) cos(27r/r) dr, (2.94) T-+00 Jo

where / is the frequency of a vibrating particle. The power spectrum is nothing but the

vibrational density of states of a collective ionic motion.

52

CHAPTER 3

PRACTICAL IMPLEMENTATION

3.1 Treatment of Classical Potential Energy

We apply the restricted quantum MD method to both an electron plasma system and

an alkali metal system. In this section, we discuss the classical potential energy for each

system. Electon-electron, electron-ion and ion-ion interactions include the long-range

Coulomb potential. We therefore introduce the Ewald summation method to calculate

the a long range potential in a small basic simulation cell with periodic boundary con­

ditions A Born-Mayer ion-ion potential and a ion-electron pseudopotential will be also

introduced for the metal system.

3.1.1 Electron Plasma System

The electron plasma system consists of N unpolarized electrons with an uniform posi­

tively charged background. The positive background is imposed to neutralize the total

charge of the system. In path-integral MD, each electron is considered as a necklace of

P beads. The only classical potential, V2 in Ekj. (2.68), of the electron plasma system is

the Coulomb pair potential. From Eq. (2.28) and (2.49), the Coulomb potential can be

written as

(3.95)

53

where is the distance between the bead (A:) of the ith electron and

the bead (fc) of the jth electron. We illustrate the Coulomb interaction between beads

in Fig. (3.7). We note that Coulomb interaction occurs over all the pairs of beads in the

1*1

Figure 3.7: Coulomb interaction between two electrons i and j at different imainary times.

discretized necklaces. This means that the electron-electron Coulomb interaction is not

retarded in imaginary time.

In an MD simulation with the periodic boundary conditions, a selection of the

size of a basic simulation cell is limited by computing capability. .A.lso the range of

an interaction between two particles is restricted within half the size of the cell (i.e.

Tc < 1/2L). Because of these restrictions, we may neglect a critical amount of long

range interactions, such as the usual slow converging Coulomb interation, between a

particle in the basic cell and a particle in any of the image cells. There is, however,

a very elegant and well-known model, known as the Ewald summation method[21], to

accrately account for the contribution of long range interactions to the energy. Following

the method, we consider a lattice made up of ions with positive or negative charges and

shall assume that ions have a spherically symmetrical gaussian distribution, with charge

54

density at radius r proportional to rj being the Ewald parameter. The calculation

procedure of the Ewald potential have two distinct but related parts. One is computing

the potential from a structure with a gaussian charge distribution at each ion site. The

other one is the potential of a lattice of point charges with an additional gaussian charge

distribution of opposite sign superposed upon the point charges. The parameter t] is

choosen such that both potentials at reference points converge rapidly.

The original Ewald summation method has been further developed by Nijboer et a/.[55].

They generalized the summation method to the interaction having form of 1/r". This

generalized Ewald summation method has been used in the Monte Carlo simulation of

the classical one component plasma by Brush et a/.[9] and of the fermion one component

plasma by Ceperley[l2]. Brush et al. showed that the Coulomb interaction for the

electron plasma system with PBC can be written as

+ E erfc(,n,) - " (3.96)

where m = —e for all i, 0, = L^\s the volume of a cell, fc is a wave vector , and rij=ri — rj

and rij is its magnitude. The wave vector can be written with an integer vector n as

k = ^n. The usual complementary error function erfc is

erfc(x) = 1 — erf(i) = 1 ^ f e~^ dy. yT JO

The last term in Eq. (3.96) is the contribution of the positive background. In path-

integral MD simulation, we may write the corresponding potential energy to Eq. (3.96)

as

= (3.97) fc=l £=l

where the summations are performed over all beads of the electrons in the basic simula­

tion cell. Specifically, the potential energy corresponding to the Ewald summation can

55

be written as

1 / 2 f 7r |np'\ f2Tr (fc)'\

+ 5 E ( 3 . 9 8 ) «=i ^ i(^j) j=i ,=i I J

where n is a reciprocal lattice vector in units such that its components are integers. If

we use the relation the first term in the right-hand side of equation

(3.98) can be made to take a considerably more eflBcient form[71]

1st—term

With equation (3.98), the double sum over i and j in equation (3.99) converts to two

single sums, i.e. ~ N'P calculations are reduced to ~ NP calculations.

3.1.2 Alkali Metal System

The alkali metal system is composed of N ions and N electrons in a cubic basic simulation

cell. In the model, the electrons are discretized in imaginary time by P beads for each

electron and the ions are dealt in a purely classicai manner. The total electrostatic

potential energy can be divided into three parts and written as

V2 = V^-' + V^-"^ + , (3.100)

where and are the electron-electron, the electron-ion, and ion-ion

potential energies, respectively. We use the Coulomb potential for the electron-electron

interaction which has been given at Eq. (3.95).

For the ion-ion pair potential, we use the Bohn-Mayer (BM) form of potential suggested

by Fumi and Tosi[27, 81] for alkali halides to model the interaction between ions. This

56

potential includes also a Coulomb potential term. The BM potential is

A T iv-l 2

I>J J=l (3.101)

where r/j = |r/ — rj\ is the distance between an ion pair I and J, and A/j and pij are

parameters of the potential. The values of the parameters can be found at the reference

[72]. We use Au = 0.6172 xlO^° eV and pij = 0.1085 A for potassium metal. The first

term of Eq. (3.101) is the Coulomb potential and the second term of the equation is the

short range core repulsion. This short range repulsion models the interaction between

the electrons in the core of the ions.

In order to deal with the electron-ion interaction, we adopt the empty core pseudopo-

tential model[R.W. Show]. In this model, we assume that a positive ion is a conducting

sphere with radius Rc and the total charge +e. The local pseudopotential on the /th

ion from the fcth bead of an electron j is defined as

from the center of the /th ion. Fig. (3.8) shows the pseudopotential. In the figure

the p>seudopotential (a) is represented by the sum of a usual coulombic potential (d), a

repulsive p potential inside the core (b) and a constant potential inside the core (c). The

total potential on the /th ion then becomes

where d[j is equai to 1 for r/j < R^, otherwise 0. We choose the core radius as Rc =

2.22 A[38]. The electron-ion potential energy can be calculated by V(''/)-

To optimize the calculation, we do not use the Ewald summation but simply replace

the long range Coulomb potentials in and by a faster converging

-e/Rc, if < Rc

(3.102)

- e / r \ y , i f r \ y > R c

( k\ |r/ — V j I is the distance of the Arth bezid of the electron j measured where r

57

(•) (b) (e)

R e

Figure 3.8: Empty core pseudopotentiaJ model for the ion-electron interaction. The sum of the potentials (b), (c), and (d) is equal to (a).

effective potential of the form

where rj is an Ewald parameter. We choose the Ewald parameter rj = 5.741/L. With

this choice of the parameter, the reciprocal space sum in the Ewald summation, the first

term of the left-hand side of Eq. (3.96), is small compared to the real space contributions

and may therefore be neglected[48]. Thus Eq. (3.104) become close to the the Ewald

summation.

3.2 Algorithm of Molecukir Dynamics

3.2.1 Overview

In this section, we will describe the overall algorithm of path-integral molecular dynamics

(MD) for both the eletron plasma system and the Alkali metal system. The MD method

developed by Alder and Wainwright (1959) is conceptually the simpler. In our model

r r ->• -erfc(T/r), (3.104)

58

system, the classical interparticle potentials are pairwise additive and central, i.e. the

total potential energy, ^ciasaicah for a system of N particles can be written as

iV N ^ciasaical — (3.105)

«=l j>i

where 0tj(r,j) is the potential between particles i and j which are separated by r,j =

|r,- — rj|. The total force on the ith particle is then calculated by

N PiiTi) = Y. /.jM' M i = 1^2,..., N (3.106)

j(><)

and

= -V>.j(ro), (3.107)

where /,j(r,j) is the force on the particle i from the particle j. We note that the potential

in this section is not including the exchange potential that will be discussed in section

3.3 with the exchange force. In an MD simulation, we compute the trajectories of a

collection of the particles in phase space which are the numerical solution of Newtonian

equations of motion. In order to set up the numerical algorithm, we start with the

Newton's force law

^ ^ 2,N. (3.108) Clu TTh

To solve the differential form of the equation, Eq. (3.108), on a computer we apply a

finite difference scheme for the second-order differential equation. We then rewrite Eq.

(3.108) as

= ^^{r.(i + A0-2r.(f)+r.(i-A«)}, (3.109)

where At is the simulation time step. From Eq. (3.108) and (3.109), we have

r.(f + A0 = 2r.(f)-r,(f-A^) + ^^^i^(A^)^ for i= 1, 2 , . . . ,N. (3.110) m

With this equation we can obtain a new position at time t + A t from the positions at

two consecutively preceding time steps and the force acting at at time t. Starting from

59

r,(i = 0) and r,(Ai) provided as an initial conditions we can compute ail subsequent

positions by appling Eq (3.110) to the system recursively. We then calculate the velocity

of the zth particle with the Euler backward scheme as

' = 1»2, (3.111)

Equations (3.110) and (3.111) is called the simple "leap-frog" integration algorithm used

by Verlet[86].

In our simulation, the number of particles, the volume and the temperature

of the system remain constants. A pair of initial configuations, (ri(0), r,(At)), are

randomly generated but subject to the constraint on the total momentun

iV m,p, = 0, af i = 0. (3.112)

:=1

where m, is the mass of the particle i . As the system develops in time the total kinetic

energy will be fluctuating due to a re-adjustment of both particle positions and velocities,

because there is a coulping between the temperature and the total kinetic energy;

= (3.113) ^ i=l ^

where T { t ) is an instantanious temperature at time t and ks is the Boltzman constant.

In order to avoid the temperature fluctuation, we rescale the velocities, u,(f + At), of

all particles in the system with a scale factor, x- For 3- constant reference temperature

Tref, the scale factor is

*<" = (3.114)

where T { t ) is obtained from Eq. (3.113). The velocity at time t + A t \ s then rescaled as

Vi{t + A«) = x(f)i;,(i), for i = 1,1,..., N. (3.115)

Algorithm Al. Overall MD-routine

1. Specify initial configuations {r°} and {r,-}.

2. Compute the velocities at time step n as

c? = (r? - r"-i)//i

60

3. Compute potentials and forces and F" with Eq. (3.106) and (3.107).

4. Compute the advanced positions at time step n+1 as

^n+i ^ 2rf - + Ff/iVm. ; ^^.(S.llO).

5. Compute the temperature at time step n as

6. Compute the scale factor x" as

x" = y/rZr^-

7. Set rf ^ r"-' and r".

8. Rescale the positions as

r, r. + x - r - ) .

9. Repeat steps from 2 to 8.

In Algorithm Al, we summarize the above calculating processes in order. We here

substituted the time step At to h. For the simulation of an ionic system, the ion reference

temperature T"" is independent to the electron reference temperature T'jfy. Thus we

determine the scale factors independently. Since the strong bead-to-bead harmonic forces

for the system with large number of bead, P, on an electron necklace may leaxl to non-

ergodic behaviors[33], the electrons are re-scaled a different way from each other. In the

step 5 of Algorithm Al, we calculate different instant temperatures for each electron

by the the summation over the beads on the same electron. The associated scale factor

for each electron is calculated by the step 6 of the algorithm.

3.2.2 Creation of the Initicd Configurations

Every MD simulation is started with a different initial configuration obtained from ran­

domly generated beads in every electron necklace. The initial bead-bead distance is

61

predetermined according to the system temperature and the expecting kinetic energy

by using the harmonic kinetic energy estimator, which is the first two terms of the left

hand-side of Eq. (2.85). We firstly generate the initial beads (r-^\ for i=l,..,N) of all

electrons with a random number generator (rani) (see the Numerical Recipes[68]. All

initial beads are generated within the basic cell. Since the electrons of our model are

unpolarized, one half (iV,) of the electrons have diflFerent spins from the other half of

the electrons. We assume that the first half and the second half of N are labeled for

the spin-up and the spin-down electrons, respectively. The following procedures are for

the spin-up electrons. Starting from the position of the first bead, about 90% of sequen-

tiai beads are generated with the predetermined constant distance in three dimensional

space. In order to reduce the time for the system to reach equilibrium from its initial

configuration, we choose that the maximum spatial extension of the necklace is equal

to 1.6 L but the necklaces remain within 0.6 L from the basic cell. The rest of the

beads are used to construct a complete closed necklace. We repeat this procedure for all

other electron necklaces and have a set of initial configuration {r|*^(t = 0)}. Another

set of configuration, {r|^'(Af)}, is created from = 0)} by generating random

displacements of the beads such that the total momentem is conserved, i.e.

r-'^'(Ai) = r-*^'(0) + for i = 1,.., and Ar = 1,.., P

with P

(3.116) Jt=i

We can use the above procedures for the spin-down electrons. Figure (3.9) shows the

initial necklace configurations for two electrons. The center square in the figure is the

basic cell. The necklaces in the figure are spreading to an image cell and they will be

broken if translational periodic boundary conditions are used. We discuss this in the

following section.

62

1-

• 'T 2STi

-r - M

Figure 3.9: Initial necklace configurations for two electrons in an electron plasma at T = 1300K and r, = 5. Only 1/3 of the beads (P = 480) denoted by filled circles are shown in xy-plain.

3.3 Periodic Boundary Conditions in Path Integral MD

The electron necklaces does not always remcdn confined in the basic simulation cell

but spread over its neighboring image cells. When some beads of an electron necklace

expand to the image cell (or cells), the continuity of the necklace is broken by the wall

(or walls) of the simulation cell. Assume that two beads belonging to the same necklace

of the electron i are at and within the simulation cell. Let us suppose that

bead (A: +1) moves outside the simulation cell in the direction (1). then its coordinates

within the simulation cell becomes — (Li,0,0), where Li is the length of the cell

in the direction (1). Periodic boundary has broken the continuity of the necklace. If we

calculate the distance between beads k and A: + 1 by the usual MD method where the

distances are calculated without maintaining the necklace continuity, we can not have

the true value of the distance between two consecutive beads. In Fig. 3.10 we show the

effect of PBC on the continuity of the necklace for P =4. We assumed that beads (1,

2, 3, and 4) make a complete necklace. If we calculate the distance between the beads

63

i

1

i 2

(a) ! 9"

i :

3 O-

<

i

1 2* V

» Q

i i

>4 ^ ' i i___i4

* L »-i

t 1

L

O

r C-1---0

I I

.__04 '

(b)

Figure 3.10: Beaxis of an electron necklace in the simulation cell and the image ceils. Beads (1 and 4) are in the simulation cell and beads (2 and 3) which belong to the same necklace are in the image cell. In the usual MD method, we use the filled circles in (a) to calculate distances between beads. In (b), the necklace is reconstructed after translating beads (2 and 3) by L from the left

64

within the simulation cell, for instance between the filled circles in (a) of the figure, the

calculated distance between 1 and 2' is not equal to the true physical distance between

beads 1 and 2. In our simulation, the position of a bead is denoted by the position vector

in the simulation cell and the address of the image cell where the bead is located in order

to allow for the reconstructing of the complete necklace. In Fig. 3.10 (b), we reconstruct

the necklace by translating beads 2 and 3 from 2' and 3' by -L, respectively. We note

that the reconstructed necklace must be counted only once in the simulation cell. If one

necklace, like the necklace of the filled circles of figure 3.10 (b), is considered as the one

in the simulation cell, then the other identical necklaces, like the necklace of the unfilled

circles of the figure, should be ciddressed to a neighbor cell.

PracticaJly, in the MD simulation, the position of the fcth bead of the ith electron is

denoted by and where is always defined inside the simulation cell and ( k )

XJ contains the information necessary for reconstructing the integral necklace of the

electron. In other words, a set k=l,2,..,P} represents a complete closed

necklace of the electron i. When we measure the distance between two beads of the same

electron i, we calculate

However if one asks about the distance between two beads in different electrons with

PBC, it does not become an obvious question. The difficulty arises because there is no

reference point for the distance measurement. In Fig. 3.11, we show the beads of two

electrons i and j, and their images. The larger circles represent the beads of the electron

i and the smaller filled dots the bead / for the electron j.

If we define a reference point as -f- then we can easily calculate the

distance from this point to the ith bead of any other electrons which can be in any of

27 cells. For a example, if we want to measure the minimum distance between the Arth

bead of the i electron and the /th bead of the j electron in any cell, we can calculate

(3.117)

- XS'^) - (rf + n„uL), (3.118)

where ticeii is an integer vector.

65

I* 1 •

elej

1 •

o

6 \ 6

P

\ 6 o

o

I • I*

ele J

. •

o

ele i

0-

I- o' elei

o

o

• 1

•

1

•

1

o

d

6

d

6

o

o

Figure 3.11: PBC diagram with two electrons. The smaller filled dots represent the /th beads of the electron j and the larger circles represent the herds of of the electron i. We assume that the beads denoted by the large filled circles are in the simulation cell.

66

3.4 Calculation of Quantum Effects

3.4.1 Evaluation of det(£'^'''')) of the Effective Exchetnge potential with PBC

When we evaluate the exchange potential and the forces, we encounter some practical

and fundamental difficulties. One of the difficulties is associated with the number of op­

erations to calculate the quantum non-local exchange effective potential. Since periodic

boundary conditions axe imposed on the system, the amount of operations dramatically

increases. For a simulation in 3 dimensional space, there are actually 26 image cells

which are the exact copies of the basic simulation cell. Fortunately, both the formulas of

the effective potential and the corresponding force have symmetric forms and have nat-

ually parallelizable forms with respect to the number of beads, P. These two facts may

be used to optimize the aigorithm. First we discuss optimization of the calculation of

the exchange effective potential. The parallel computational algorithm will be discussed

in a later section.

In order to discuss the algorithm for the calculation of the effective potential with

the path integral molecular dynamics, let us rewrite the effective potential of N'-iso-spm

electron system:

. (3.119)

where

K57" = f;E^(r!''-••!»«>)= (3.120)

and

= "i E E K,- (3-121) ^ p=i,=i ^

In the above equations, and are the harmonic potential and the exchange

potential functions, respectively. A matrix element (£'^''''^),j is given by

[-^ {(.(-I - - r!-')^}] (3.122)

In the above equations, the subscript and the superscript are used for the index of

electrons and the index of node of a given electron, respectively. With the periodic

67

boundary conditions in 3 dimentions, the system is constituted of a basic simulation cell

and 26 image cells. As the usual molecular dynamics, an electron in the simulation cell

interacts with all other electrons in the simulation cell and the neighboring cells. Then

the exchange process must be considered between an electron in the simulation cell and

any electron in the neighbor cell as well as exchange process between two electrons in the

simulation cell. We, therefore, have to expand the size of matrix to (27iV' x27N')

instead of (N' x N'). For instance, if one has 27 spin-up and 27 spin-down electrons in

the simulation cell (like in our metal system), the size of the matrix will be (729 x 729).

It is practically impossible to calculate efficiently determinants of this size. To find a

good approximation, let's expand the determinant as the following:

N' N' jV iV N' det (£(-•'1) = I - Z E E f i r ' + E E E E sS"'

cells t=l j=l cells t=l j=l Jfc=l

N' N' N' N' -EEEEE' '1 ;« ' + - - (3 -123 )

cells »=1 J= l /= l

where the summation JZce/Za means that every index of particles (i.e. i, j, k and /, ..) is

running over the all 27 cells. 1 arises from the diagonal term of the determinant, because

all diagonal terms of the matrix are 1, and /,j, gijk, and hijki represent 2-cycle, 3-cycle,

and 4-cycle exchange processes, respectively. Since we are interested in calculationg

exchange force acting on the electrons within the simulation cell, we can restrict the ith

electron within the simulation cell so that the index of the tth electron in Eq. (3.123) is

independent to the summation over cells. In other words, the size of the matrix

is reduced to (27iV' x N'). To gain some intuition for this expansion, the second term of

the equation which corresponds to exchange between pairs of electrons may be written

as

E E fir-' = E E exp [-1^ {(r!" - r f f - (r!-l -cells i,j=l cells i.j=l I

+(rl-' - r!")2 - (r;»l -

N' N'

cells t=l J=l

68

,V' f iV .V N' 1 = E E + E + • • •+E [ • (3-124)

1=1 ij=i j=i j=i )

where

cp = _ r(''))2 'J > '

4 = (3.125)

L^f = 4 + 4 _ _ ^2^ R- = 1,2, • - •, 27. (3.126)

The index i in the above equations is restricted to electrons within the simulation cell,

but the index j can point to any electron over all neighbor cells including the simulation

cell itself. Figure [ ] shows two-cycle exchanges between a pair of electrons. One of the

pair of electrons should be kept within the simulation cell.By choosing a minimum value

of Llf^' over all K, we have

N' N' N' N'

EEE/r ' -EE ' - '™" cells i=l j=l i=l j=l

V \ (3.127) I Kl^Ko) )

The second term in the above equation is much smaller than I for most configura-

tions.Thus we can make the approximation:

1 ' N' iV iV' iV N' ;V' N'

E E E ft' = E E = E E E E (s-^s) cells i=l j=l i=l _7=l :=lj=lt=lj=l

Furthermore, we can contineously apply this approximation to higher cycles of exchange

processes. Then we may have that

^ 1 - E E +E E E sisri. - • • • (3129) i=l j=l t=l j=l fc=l

or

det(£J27yv'x27iV') ~ (3.130)

This approximation , which includes all possible exchange cycles, works better than the

weighted 2-cycle approxmation suggested by Hall.

We details the practical computing algorithm at fiqure 3.12.

69

AJgorithm 2 Calculaing det( *).

1. for i = 1 to P 2. for j = i to P

3. for ie = 1 to N-1 4. for je = ie to N

Calculate dl = [(r\,+X\,-Xl)-r',S-

5. for cell = 1 to 27 Calculate

rf3(ce//) = ({rU + X;. - X{^) - (r]^+n„nL)f d4{cell) = Hr), + X], - Xj.) - (rf, + n„uL)? arg(ceU) = \d3(cell) + d4{cell) -dl- d2\

6. Find the cell number. Cm, such that arg{ceU) has a minimum at the cell and set

argl = —Co(rf3(cm) — di] and arg2 = —Co(t/4(Cm) — d2) arol

cU.') — g-argt je.»e ^

7. Repeat 4 and then 3.

8. Put 1 to all diagonal elements of and Calculate det(£^''*') by using the LU-decocomposition algorithm. Evaluate det(£« *>),

if det(£'^' *') > 0, calculate exchange force and then repeat 2 if det(E<' **) < 0, repeat 2.

9. Repeat 1.

We use that Co =

Figure 3.12: Algorithm for the calculation of the determinant det(E(''-'^).

70

3.4.2 Effective Force Calculation

To perform the molecular dynamics, we have to calculate force on each particle. The

effective force originating from the effective quantum potential is given as

f e / f = fk arm "t" f exch

where fiuirm. f exch the harmonic force and the exchange force, respectively. Let's

consider the effective force on the bead (Ar) of an electron (t). The harmonic force may

be written as

{(-!" - - c!*-" - -l")} (3.131)

The harmonic force is calculated with a closed necklace as shown in Fig. 3.10 (b). The

exchange force can be derived by calculating of ^exch with the following

matrix algebra:

n

det.4 = ^aij Aij, i=l

«=l j=l

where A is an ( n x n ) square matrix, a,j is an element of matrix .4, and A,j is a cofector

of the element a,j. The exchange force on the bead (k) of an electron (i) is given as

where is a cofactor of a matrix element In odrer to solve this equation,

we explain the details at APPENDIX C. The exchange force is

= (1^) f (3.133)

and the elements of matrix and are

(rl"'_rS»')(£<'•»))„ if p=i

if p i (3.134)

71

and

( > • ! . » ' - i f q = i (3.135)

i f q ^ i .

During the calculation of the exchange forces, we choose configurations which have a

positive determinant value. .\s we mentioned in section 3.3, a necklace of an electron

must have a closed form and be treated exactly in the same manner both in the harmonic

force and the exchange force calculations.

3.5 Parallel Computation

In the path-integral molecular dynamics of a many-fermion system, most of computing

time ( > 95 % of total time) is devoted to the calculation of the exchange force and energy

of a particle. We have shown in the previous section, that the calculation of the exchange

force can be accelerated by making appropriate approximatins. In this section, we show

that the exchange forces calculation can be implemented on the parallel computer leading

to a significant reduction in computing time.

One often wirtes the program with vectorization algorithm to use a vector-type

computer for a long computational job. The vector-type machine, however, can not be

very useful for this particular simulation, because the inner most routine of our simulation

contains a determinant calculation which can not be vectorized. One other method to

reduce the real computing time (i.e. the wall clock time) for a long simulation is a parallel

computation. Our simlulation algorithm is an ideally sited for the parallel computation,

because we can independently evaluate the exchange forces on P different nodes of an

electron necklace. If we have an M-processor computer and assign P/M jobs to each

process, we can finish the job M times faster.

To see the parallel algorithm clearly, let's consider a physical quantity A

P

(3.136) 1=1

72

where can be the exchange force or the exchange energy of a single bead i of an

electron necklace. Because the caJculation of can be performed independently of the

calculation of any other equation (3.136) can be rewritten as

A = x: + • • •+e A h Im

or M P'

.4= (3.137) 1=1 t=i

where P' = P / M , J [ i ) = i" + (/ — 1)M, and M is a positive integer which is equal to

the number of processors. With equation( 3.137), each single processor only calculates

rather than in a normal serial computer. Figure 3.13 shows

the sequantial order of the parallel computation. First of all, we allocate M different sub-

jobs, to M processors with ail information necessary for the calculation delivered

by a processor which is called the Master. Upon complete reception of the necessary

information each processor sends a signal to the Master and starts starts to calculate

the subjob. When the processor has finished the subjob, it sends send the results to

the Master followed by a job-end signal. When the Master gathers all job-end signals

with the results from the M processors, the Master proceeds to a next sequantial job.

The allocating and the gathering procedures of information takes extra computing time

which is called the communication time. In our simulation, the total amount of commu­

nicating data is normally on the order of 1 Mbytes. No data have to be transfered from

one process to another during the calculation. Since the bandwidth of a communication

network is about 10 Mbytes/sec, the communication time is less than 1 % of the com­

putational time. In figure (3.14), we illustrate the scalability of computing time by the

number of processors. The computing speed has been increased very sharply up to 30

processors.

To implement the parallel program, we use the Message-Passing Interface (MPI)

[31] library tools which is based on the message-passing model machine. The message-

passing model posits a set of processes with local memory which are able to communicate

with each other processes by sending and receiving message through a communication

73

PO(Master): allocating all information

to M slaves

PO: gathering all results

from the slaves

Figure 3.13: Sequantial job order in a parallel computer.

74

1-0

0.8

0.6

0.4

0.2

0.0 20 40 80 100 120 0 140

Number of Processors

Figure 3.14: Scalability of the parallel calculation. CPU time scale is normalized to 1 for the seriaJ calculation with one processor. All time is measured at the IBM SP2 in the Telecommunication Center at Cornell University.

75

network. MPI is a specific realization of the message-passing model. IBM SP2(in the

Telecommunication Center at Cornell University) and SGI Origin 2000(in the CCIT at

University of Arizona) have been used to run the parallel simulations.

76

CHAPTER 4

APPLICATIONS AND RESULTS

4.1 Electron Pkisma

4.1.1 Model System

We have tested the restricted path integral molecular dynamics on an unpolarized elec­

tron plasma composed of NeU = 30 electrons(iV, = 15 with s = spin-up or spin-down).

.A.n uniform positively charged backgroud is imposed to neutralize the system. The sim­

ulation cell is a fixed cubic box with edge length £.=13.3A, 19.95A, or 26.6A, which

correspond to electronic densities with 7.5, and 10, respectively. r,=r/ao where

Bohr radius and r = (3/47r7z)'''^ is the electron radius for the electron number density n

and oq is the Bohr radius. We applied periodic boundary conditions in all three dimen-

tions so that the system is constitutied of one basic simulation cell and 26 image cells.

Under these conditions the total potential energy is equal to the sum of the interactions

within the basic cell plus the interactions between the basic cell and all 26 image cells

minus a background term.

With Eq. (2.80) and (3.98), we solve the equation of motion with a leap frog

scheme and an integration time step At = 2.010"^® sec. Most simulations were run for

an average of 30,000 time steps. In some cases for the low and intermediate density

plasmas, we have run simulations up to 50,000 steps for better equilibration.

Because of the large computational cost of the calculation of the exchange effective

potential and forces, the exchange forces are calculated and updated every 10 time

I I

steps. The values for the exchange forces are used subsequently during the 10 time

steps following their calculation. We have compared the average energies obtained one

from the sicip-procedure and the other from the non-skip-procedure, found no significant

statistical difference in their values. We show the energies obtained from both procedures

in Fig. 4.15.

The chosen time step is small enough to resolve the high frequency oscillations

of the harmonic motions related to the potential. In the case of system with large

P, the strong harmonic forces in equation (2.60) may lead to non-ergodic behavior[33].

This problem can be alleviated by rescaling temperature with a necklace of Nos^Hoover

thermostats[49, 50]. This rescaiing would ensure convergence to the right canonical

distribution. We have elected to rescale the temperature of each necklace of P beads

independently of each other via a simple momentum rescaling thermostat[93]. With this

procedure we do not obtain a true canonical distribution, but most thermal averages will

be accurate to orders N~^ [30].

We chose an Ewald parameter t} = for which satisfactory convergence is

obtained with truncation of the real space sum at Z,/2 and truncation of the reciprocal

sum at < 49. As described in section (3.3), every simulation starts with independent

initial configuration obtained from randomly generated bead positions in every electron

necklace. The kinetic energy is calculated with the first two terms of the energy estimator,

Eq. (2.85), because {KEexch) is very small and stable values ( -0.05 ± 0.01 eV/electron

for r, = 5 aX T = 1300K and 1800K) and requires more than twice of computing time

necessary for the exchange force calculation. We also note that the error of the kinetic

energy estimator has been estimated by calculating the standard deviation on the running

cumulative average over the last 15,000 time steps of each simulation. This error is on

the order of 0.03 eV per electron.

78

170

a

§ 150 c 3 •a 0 u 3

1 130

110 4000 2000

Time step 1000 3000

3.0 o

c 2.5 3

- 2.0 3 "O &

O) w 1.5 o c o

1.0 c o o Q.

0.5 0 2000 1000 4000 3000

Time step

Figure 4.15: and potential energy in reduced units as functions of time steps. In both cases, the thick lines and the thin dotted lines refer to skip = 10 and skip = 0, respectively.

79

4.1.2 Results

First of all, we have investigated the convergence of the energies with respect to the

number of beads, P, in an electron necklace. In figure 4.16, we report the kinetic energy

of the high density plasma (r, = 5) at T = 1800 K and the intermediate density plasma

(r, = 7.5) at r = 700 K as a function of the number of beads. The convergence of

the potential energy with respect to P is illustrated in figure 4.17 for r, = 5 at T" =

1800 K and r, = 7.5 at T" = 700 K cases. We note that the energies converge to

some asymtotic value for necklaces containing as few as 200 to 300 beads even for the

electron plasma near metallic density. This observation is particularly significant as the

non-parallelized path-integraJ molecular dynamics algorithm scales with the square of

the number of beads. In additio, to PIMD energies, we have also indicated the 0 K

kinetic and potential energies of electron plasma with same density of reference 16. .A.t

the temperature of 1800 K and 700 K, the high and intermediate density systems are

in the degenerate regime and the electronic energies should be comparable to the T =

0 K values. The kinetic energies are in very good agreement but some discrepencies

e.xist between the potential energies as the restricted PIMD appears to over-estimate

them. In order to further the validation of the restricted PIMD, we have conducted a

series of calculations at several temperatures for the three densities. For the high and

medium density systems we have used 450 and 300 beads, respectively. These number

of beads fall within the region of convergence. Electrons in the low density electron

plasma are discretized over 360, 380, 450, 680, 720, 780 for the temperatures 1100, 900,

450, 575 and 550, 500 and 450, 400 and 350 K. The calculated kinetic energies of figure

4.18 are in excellent agreement with the variational Monte Carlo results of Ceperley [12]

for correlated one-component plasma. We note that the kinetic energy is not varying

significantly over the range of temperature studies as is expected for these plasma at the

border of the degenerate and the semi-degenerate regimes[20]. At low temperature, the

low density system with large numbers of beads takes a very long time to equilibrate

and sampling of phase space is not very efficient. In this case, calculation of reliable

energies require very long simulations. Another difficulty in calculating reliable energies

80

I I f 1 T

200 400 600 800 Number of Beads (P)

Figure 4.16: Kinetic energy of electron plasmas as function of number of beads in the necklace representation of quantum particles. The circles and squares refer to the high density (r, = 5, 7'=1800K) and medium density (r, = 7.5, r=700K).

81

-1.0

200 400 600 Number of Beads (P)

800

Figure 4.17: Potential energy as functions of number of beads. The circles and squares refer to the high density (r, = 5, r=1800K) and medium density (r, = 7.5, r=700K).

82

when large number of bead are used results from the fact, as was noted before, that

the variance of the kinetic energy increases with P. We did not need to use so many

beads for the low density plasma even at low temperature, however, these simulations

illustrate the need to use as few beads as possible within the interval of convergence. In

addition to the T" = 0 K correlated energies, we have indicated the Hartree kinetic energy

(2.21/r^ in Rydberg) with a dotted line. Fig. 4.18 shows that the non-local form of the

density matrix given by equation (2.59) introduces some electron correlation. This is

also apparent in the results for the temperature dependence of the potential energy. The

calculated potential energy falls between the fully correlated results of Ceperley and the

electron-electron interaction contribution to the Hartree-Fock energy (given by -0.916/r,

in Rydberg). We also note that the potential energy increases weakly with temperature

and that extrapolation toward T = 0 K should result in potential energies in better

agreement with the correlated potential energies than uncorrected ones. In the present

model, however, the non-local effective potential introduces electron correlation between

electrons with identical spins only. The present potential energies are over-estimated as

correlations between electrons with opposite spins are not ciccounted for. In (B) of the

table we linearly fit the potential energies by using a -I- bT. In table 4.1, we show the

kinetic and potential energies of different models.

Finaily as a demonstration of the effectiveness of the exchange potential in equa­

tion (2.73), the pair correlation foriso-spin and hetero-spin electrons is reported in figure

4.19 in the case of high density plasma at the three temperatures studied. The difference

between iso-spin and hetero-spin radial distributions is striking. In order to satisry Pauli

exclusion principle, the non-local exchange potential keeps the electrons with identical

spin away from each other while electrons with different spins can approach each other

quite closely. The Coulomb repulsive force is the only force keeping electrons with dif­

ferent spins from approaching. The non-local exchange potential is quite short range

as it does not appear to affect the electron distribution beyond 5 A. The exclusion is

particularly important in the interval (0, 3 A). The major effect of a rising temperature

is the increase in pair correlation at shorter distance or in other words, the shrinkage of

83

(A). Kinetic energies per electron.

r,(= r / a o ) Current model (eV)

reference 12 (eV)

Haxtree-Fock Approximation(eV)

5.0 1.511 (± 0.045) 1.529 1.203

7.5 0.764 (± 0.022) 0.774 0.534

10.0 0.523 (± 0.076) 0.491 0.301

(B). Potential energies per electron.

r,(= r/ao) Current a(eV)

model b (lO-* eV/K)

reference 12 (eV)

Haxtree-Fock Approx.(eV)

5.0 -3.396 ±0.1367 2.73 -3.619 -2.493

7.5 -2.355 ±0.0306 1.51 -2.517 -1.662

10.0 -1.685 ±0.0474 1.46 -1.944 1.246

Table 4.1: Kinetic (A) and potential (B) energies per electron for various electron den­sities. Kinetic energies per electron of current model are the average values. Potential energies per electron are linearly fitted by using PE = a + bT.

84

2.0 • I > 1 ' ' 1 r—r—I 1 1 1 1 r

0 500 1000 1500 2000 2500 Temperature (K)

Figure 4.18: Kinetic energy as functions of temperature. The electron plasma with ra=5, r,=7.5 and r5=10 are refered to by circles, squares and triangles, respectively. The horizontal thick dashed lines correspond to the energies of Ceperley[12,13]The thin dotted lines indicate the Hartree-Fock energies.

85

-1.0

-1.5

-2.0

-2.5

-3.0

-3.5

-4.0

» !

I

' •

500 1000 1500 Temperature (K)

2000 2500

Figure 4.19: Potential energy as functions of temperature. The electron plasma with ^3=0, rs=7.5 and rs=10 are refered to by circles, squares and triangles, respectively. The horizontal thick dashed lines correspond to the energies of Ceperley[12,13]. The thin dotted lines indicate the Hartree-Fock energies. For both types of lines, r,=5, r,=7.5 and r,=10 are represented from the top to the bottom.

86

1.0

hetero-spfn c o

W (0

a 0.5 0 1 o

I

I

iso-spin r :

0.0

Radial distance (Angstrom)

Figure 4.20: Iso-spin and hetero-spin electron-electron pair distributions for the high density (r, = 5) electron plasma at T=1300 K (solid lines), T=1800 K (dotted lines) and T=2300 K (dashed lines).

87

the exchange-correlation hole[20].

After showing that the non-local restricted PIMD can simulate with reasonable

accuracy electron plasma near metal density, we apply the method to the simulation of

the solid and liquid phases of an alkali metal from first-principle.

4.2 Soiid/Liquid Transition of Alkali Metal

4.2.1 Model System

We simulate crystalline and liquid potassium metal with method of the restricted path-

integral molecular dynamics. Potassium has been choosen because

(1) it is a prototype free-electron metal which has been studied previously by semi-

empirical pair potential,

(2) there exist experimental data for the pair-correlation function of the liquid state[89],

thermodynamic[3] and vibrational properties[19].

The alkali metal system is composed of 54 potassium ions (K"*") and 54 unpolarized

electrons in a cubic basic cell with edge length L. The number of electrons with spin-

up and spin-down is N-a,-p = 27 and N^onm = 27, respectively. In the solid phase, the

potassium ions and the electrons are arranged on a body centered cubic (bcc) lattice.

We start every simulation with a newly generated random configuration. In contrast

to the electron plasmas, when ion temperature is varied, the dimensions of the basic

cell are adjusted to match the experimental density of K crystal. The density of K

crystal is linearly varying with temperature; D{T) = Dq+ [T — Tojo:, where Dq = 0.827

gjcrr? is the density at the melting point Tq = 337 K and the expansion coefficient a = -

0.2285 x].Q~^g/Kcm^ [76]. Unexpectedly, our simulation have shown that the potassium

model system melts at a temperature below the experimental vaJue of the melting point,

consequently the density of the liquid system reported below conforms to the value of

the density of crystal.

88

Periodic boundary conditions (PBC) are applied in three dimensions to the basic

simulation ceil and the cutoff range of potentiaJ. is chosen to the half size of the basic

cell, i.e. £/2. We use Au = 0.6172 x 10^° eV and pu = 0.1085 A for the Born-Mayer

potential for the ion-ion interation, Eq. (3.101) [72], a core radius Rc = 2.22 A [38] for

the ion-electron pseudopotential, and the Ewald parameter 77 = 5.741/Lo . I.0 = 16 .4..

The physical potassium ion mass is Mi = 71,830 me, leading to an extreme

disparity in electronic and ionic time scales. For practical reasons, we use a ratio of

the ion mass M/ to the electron bead artificial mass, m*. of 39.1 ; 1, i.e. m* = 1

amu. The dynamics of the electrons is still significantly faster than the dynamics of the

ions. We solve the equations of motion with a leap frog scheme and an integration time

step At = 2.8 X 10"'® sec. Most simulations were run for a minimum of 70,000 time

steps (~ 20 psec). In some cases for the calculation of vibrational properties, we have

run simulations upto 130,000 steps. As we described in section 4.1.1 for the simulation

of the electron plasma, the exchange forces are calculated and updated every 10 time

steps. Then we use the updated exchange forces for the next 10 time stef>s to reduce the

computational cost.

We have studied the potassium model system at temperatures in the interval

[lOK, 298K]. The simulation of the electronic degrees of freedom as discrete necklaces

as these low temperatures would necessitate a large number of beads for convergence

with respect to P. From the study of an electron plasma at high density (near the metal

density of potassium), the electron systems conserves a nearly degenerate character up

to a temperature of 2300K. Since temperature does not affect significantly the electron

states at the metal density, we have thermally decoupled the classical ionic degrees

of freedon and the quantum elctronic degrees of freedon. The electron necklaces are

attached to a thermostat set as a temperature of 1300K while the ionic temperature

is adjusted independently with another thermostat. At the electron temperature of

1300K, it is sufficient to employ a reasonably small number of beads for convergence of

the algorithm.

89

The calculation of the electron kinetic energy is done in a similar fashion to that

of the electron plasma. With the kinetic energy estimator, the kinetic energy is given as

a small quantity, difference between two larger quantities, with a variance growing with

P. This estimator, therefore, introduces an error on the calculated values of the average

kinetic energy. We have estimated this error by calculating the standard deviation on

the running cumulative average over the last 30,000 time steps of the simulations. This

error is estimated to be on the order of 0.01 eV per electron. Figure 4.21 shows the

running averages of the electron kinetic energies at T = 273K and T = lOK.

4.2.2 Results and Discussion

In a first stage, we have investigated the convergence of the algorithm with respect to

the number of beads in the electron necklaces, namely P. For this we have calculated

the electron kinetic energy at an ion temperature of 273A' and an electron temperature

of 1300A' for systems with varying values of P. It is important to note again that each

simulation starts from different initial necklace configulations. Figure 4.22 presents the

results of these calsulations. It is clearly seen that the electron kinetic energy converges to

an asymptotic valuse of approximately 1.23 eV/electron. The algorithm appears to have

nearly converged for number of beads exceeding 240. As a trade off between accuracy

and efficiency, we have choosen P=260 for all subsequent simulations.

The total energy of the potassium system as a function of temperature is reported

in figure 4.23. The energy shows two regions separated by an apparent discontinuity of

approximately 0.025 eV/atom. In figure 4.23, we have also drawn as a guide for the

eyes best second order and first order polynomials fits to the low temperature and high

temperature energies, respectively. The slope of the curve increases significantly from

the low to the high temperature region indicative of larger energy fluctuations in the

high temperature systems. In table 4.2, we compare the slope of Fig. 4.23, i .e. AE/AT,

and the gap of the total energy at T = 230K with the experimental values.

90

(A). Specipic heat

SE/ST (Current model) C„ (experiment) (10""* eV/atom K) (10""* eV/atom K)

Solid 1.810 ± 0.0145 2.540^^^

Liquid .3.786 ± 0.0144 3.048(2)

(B). Latent heat of fusion

AE at 230K ~ 0.025 eV/atom

AH (experiment) 0.0245 eV/atom

Table 4.2: Specific heat of the solid and the liquid, and latent heat. The experimental values are at the data book(CRC). (1) and (2) in (.A.) are measure at T = lOOK and 298K.

It shows that the vibrational modes of the model metal are softer than their

experimental counterparts.

We recall that the density of the simulated potassium system varies continuously

as a function of temperature as it is set to the temperature dependent density of the

solid. Therefore the discontinuity is not associated with any discontinuous change in vol­

ume of the system but can only result from a structural transformation. This structural

transformation takes place around 210A'. .\s we will see later from structural data, this

is a solid to liquid transformation. The calculated transformation therefore underesti­

mates the melting point by nearly 12GR' as potassium melts at SSS/C under atmospheric

pressure. This difference cannot be assigned to the fact that the density of the simulated

system is constrained since such a constraint should have the opposite effect of raising

the melting point. The difference between experimental and simulation melting point

can only result from the computer model that underestimates the strength of the K-K

bond and in particular we believe that it is a consequence in part of the approximation

made to reduce the range of the Coulomb interaction. In that respect it is predominantly

a size effect.

91

To gain further insight into the energetics of the transformation, we have graphed

in figure 4.24, some of the contributions to the energy of the system. The only energy

that is not plotted is the classical kinetic energy of the ions. Since the temperature of

the ions is maintained constant by a thermostat, the ion kinetic energy is a simple linear

function of temperature and cannot account for the discontinuity in the total energy.

.\part from an isolated point at 200A', the potential energies vary reasonably continu­

ously with temperature. In contrast, it appears that the electron kinetic energy data

is separable in two groups, namely a low temperature group and a high temperature

group. Since the electrons in the potassium system are nearly degenerate, their kinetic

energy should not be temperature dependent provided the atomic structure remains the

same. Within each group the kinetic energy does not show any systematic variation. We

should remember that the standard deviation on the electron kinetic is approximately

0.01 eV/electron. The difference between the energies of the two groups amounts to

approximately 0.015-0.02 eV/electron and appears to be a significant contribution to

the total energy discontinuity. The raise in kinetic energy as one crosses the disconti­

nuity from the low temperature to the high temperature is indicative of a change to an

electronic state of higher localization in the high temperature metal. This observation is

consistent with the expected behavior of electrons in a liquid structure in contrast to a

crystalline solid. As the structure disorders from crystalline to liquid, one anticipates a

narrowing of the electronic band. However, since the short-range local atomic environ­

ment does not chanfe drastically between the liquid and the solid above and below the

transformation temperature, the e.xtent of the electronic localization should be small.

We characterize the atomic structure of the simulated system via the ion-ion pair

distribution function. The distributions calculated at several temperatures are drawn in

figure 4.25. The very low-temperature ion-ion pair distributioni function shows a first

nearest neighbor peak at approximately 4.6A and well-defined second nearest shoulder

at 5.3 A. The third nearest neighbor peak occurs near 7.4 .4. These features are char­

acteristic of the body centered cubic structure of crystalline potassium. As temperature

increases, the second nearest neighbor shoulder fades away and merges with the first

92

nearest neighbor peak forming a broad asymmetric peak because of the large amplitude

of atomic motion. At the temperature of 76A', the third nearest neighbor peak retains

its identity. At 150^, this peak consists only of a vague shoulder part of a much broader

peak that should encompass higher order nearest neighbors. However, due to the limited

size of our simulation cell, we cannot resolve with much confidence the pair distribution

function beyond one half the length of the edge of the simulation cell. On the same

figure, we have also plotted the ion-ion pair distribution functions at the temperatures

of 248A" and 273K. The distributions at 273A' and 298A' are practically identical. The

maximum of the first nearest neighbor peak shifts toward lower values as temperature

increases. At 273A", this maximum occurs at a distance of approximately 4.3 .4. This

distance is an underestimation of the experimental first nearest neighbor distance of the

ion-ion distribution of potassium [89] but the calculated liquid distribution is in good

qualitative accord with available experimental data.

To supplement the structural information provided by the ion pair distribution

functions, we report in figure 4.26 two-dimentional projections of the trajectories of

the potassium ions at several temperatures. The first two figures,4.26 (a) and 4.26

(b), correspond to the crystalline states. The ionic species vibrate about well defined

equilibrium lattice positions. At the two high temperatures, fig. 4.26 (c) and 4.26

(d), one cannot identify lattice positions anymore. Although one may still identify

some vibrational component to the ionic motion in the form of some localization in the

trajectories, ionic motion is not predominantly oscillatory but also possesses a diffusive

charater.

More quantitative information concerning ionic motion is available from the

analysis of the mean square displacement. Figure 4.27 shows the mean square displace-

ment(MSD) of potassium ions as a function of time and temperature. In terms of the

MSD, diffusive motion is identified by linear variation with time in the limit of large

time[5l]. Vibration motion is characterized by a time independent MSD. At the three

lowest temperatures(10, 76, and 150A), the MSD indicates that ionic motion is vibra­

tional. At the highest temperatures of 248, 273 and 298K, the ions exhibit diffusive

93

motions. It is somewhat more diflScuIt with the temperature of 223K. However, because

the density of the system is constrained to conform to that of the solid, it is not surpris­

ing that at 22'3K, atoms in this liquid may display essentially vibrational motion. We

also illustrate the vibrational amplitudes of ions as a function of the ion temperature in

figure 4.28. The vibrational amplitude is taken as the maximum amplitude experienced

by an individual ion during its trajectory. Individual amplitudes are then avetaged over

the ions and reported in figure 4.28. In this figure, the vibrational amplitudes of ions are

almost linearly increasing up to T =223K. The slope of the dashed line is 6.33 xlO~^

A/K. In this low temperature region, the increase in amplitude may be associated with

an increase in the number of phonons. A simple dimensional analysis predicts that the

amplitude increases as y/T in classical temperature regime. Because the data of figure

4.28 are so qualitative one cannot predict accurately the actual temperature dependence.

At T = 250K, we can still estimate a vibrational amplitude owing to the partial vibra­

tional nature of ionic motion. This amplitude exceed the crystal value by as much as a

factor of 2, showing a structive transformation.

Further information on the ionic motion is obtained from the calculation of the

normalized velocity auto-correlation function(NV.A.F). We also consider the power spec­

trum of the NVAF. defined as its Fourier transform. The NVAF's and associated power

spectra have to be analyzed in a qualitative manner because the time over which they

are calculated is not long enough for quantitative characterization. In figures 4.29, the

NVAFs at the two high temperatures of 21ZK and 298/v" show features of the crystalline

state such as oscillations representative of thermally excited phonons in crystal lattices.

The contrast in ionic motion between the liquid and the crystalline is also quite apparent

in the power spectrum and in particular in the low frequency modes. At lOA', the power

spectrum drops to zero at zero frequency. The liquid systems at 273A' and more evi­

dently at 298A', exhibit non-zero values of the power spectrum at zero frequency. This

observation is in accord with a diflFusive ionic motion [78], 2044(1992)]. The peaks in the

power spectra of the liquid metal are consequences of the oscillations in the NVAF's and

may thus be regarded as remnants of the phonon structure observed in the crystal state.

94

The fact that the density of liquid is constrained to that of the crystal may accentuate

this effect. .As temperature increases or density decreases, these peaks should disappear

with the decay of the oscillations. It is not possible to extract detailed information from

the fine structure of the power spectra because of the finite time used in their calculation.

However, one may compare qualitatively the calculated power spectrum at lOA' with that

deduced from experimental measurements at 9 A' [19]. The experimental phonon density

of states possesses a major peak near 2.1 x Vibrations in the PEMD model of

the crystal potassium have lower frequencies in the range 0.8-1.3 x suggestive of

weaker bonds. This observation correlates closely with the observation of a calculated

melting temperature underestimating the experimental melting point.

Finally, we consider the change in electronic structure of the metal upon melting.

This change is associated with an increase in electronic kinetic energy of approzimately

0.02 eV/electron. This energy is small and thus one expects only a slight modification

of the electronic structure. Such a variation is obserable in the electron pair distribu­

tion function of figure 4.31. The partial pair distribution functions show that the major

difference between the low temperature crystal and the liquid is an increase of the max­

imum in the hetero-spin pair correlation between 3A and 4A and an ezpansion of the

exchange-correlation hole as seen in the iso-spin distribution. In a previous study of the

effect of temperature on the electron density in an electron plasma with near that of

the present potassium system, we had shown that increasing temperature shrinks the

exchange-correlation hole[58]. However, the direct effect of temperature in the electronic

dtructure cannot be a factor as it is maintained constant by a thermostat. Here, the

e.xpansion in the parallel-spin electron correlation may thus simply be a result of vol­

ume expansion. On another hand upon melting the first nearest neighbor and second

nearest neighbor shells of the crystal structure collapse and the ion coordination num­

ber in the liquid increases. The exchange-correlation force between neighboring iso-spin

electrons may then induce further localization. The resulting localization within and at

the border of the ionic core is seen best in the electron-ion radial distribution of figure

4.32. At low temperature the ion-electron pair distribution function shows a significant

95

first maximum at a distance of 2.2A. This distance corresponds to approximately one

half the first nearest neighbor interatomic distance. The ion-electron correlation reaches 9 9

a minimum between 4.3 and 4.5 A followed by a second maximum near 6.5 A. The

ion-electron pair distribution is therefore complementary of the ion-ion distribution. In

other words, high electron-ion correlation is expected where there is low ion-ion correla­

tion. The high electron density between ions is indicative of bonding. Considering now

the mid-points between ionic sites as consisting of electronic sites, we can estimate the

electron-electron distance in the potassium bcc structure to be on the order of L/\/2

times the lattice parameter. The electron-electron distance thus calculated amounts to

approximately 3.67 A. This number is in e-xcallent accord with the observed maximum in

the hetero-spin electron-electron pair distribution. With this information, we may con­

struct a simplified picture of the electronic structure in the crystal phase. The electron

density is highest between the ions thus leading to bonding and the electronic sites are

occupied alternatively by electrons with differing spin.

An increase in electron localization at the electron sites occures even in the solid

state at the higher temperature of 150A*. This shows that atomic vibrtations have a

significant effect on the nature of the electronic states in crystalline potassium. Vibra­

tions tend to localize the electron density.Similar observations by other investigators were

made for the case of sodium clusters[35, 36]. The electron density localizes further with

disordering of the structure at even higher temperature. In the liquid, the electron den­

sity increases near 2.2 A. This increase is compensated by a reduction in the electron-ion

pair correlation at longer range as seen by the loss of electron-ion correlation near 6.6 A.

Since the calculated cynamical properties support the retention of vibrational motion in

addition to diffusive motion in the liquid state, it is nuclear at this stage which of two

processes: vibration versus disorder, contributes principally to the localization.

The electron localization at the edge of the ionic cores should leaA to an increase

in hetero-spin correlation between 3 and 4 A which is observed in figure 4.31. Finally, we

note that the larger electron density between nearest neighbor ions is consistent with the

shorter ion-ion bond length in the liquid structure. In order to see the electron necklace

96

with ions, we add figure 4.30. Althought it is hard to tell anything quantitatively, we

still can find that the most beads remains between ions and spreads over several ions.

97

1.24

T = 273K

1.22

O)

.A>v f

T = 10K S 1.20

1.18 40000 60000 50000 70000

Time steps

Figure 4.21: Running averages of electron kinetic energies at T = 273K and T The standard deviations are 0.003 (eV/eiectron) and 0.005 (eV/eiectron) at T and at T = lOK, respectively.

= lOK. = 273K

98

1.40

J -Si >

>. O) h. o c o o

c o o o lU

1.20

1.00 -

0.80

I

100 150 200 250 Number of Beads (P)

300 350

Figure 4.22: Convergence of electron kinetic energy with respect to the number of beads (P) in potassium. Tion = 273 K and Teu = 1300 K. Nion = = 54.

99

50 100 150 200 250 300 350 Temperature (K)

Figure 4.23: Total energy of the potassium model versus temperature. The lines are fits to the data in the low and high temperature regions.

100

E o o > 3-O) w O C o «

o a.

TJ c a •

I •¥

-2.30

-2.35

-2.40

-2.45

-2.50 100 200 300

Temperature (K)

o 1.24

100 200 300 Temperature (K)

100 200 300 Temperature (K)

Figure 4.24: Various contributions to the total energy of the potassium system as func­tions of temperature.

101

T = 10K T = 76K T = 150K T = 248K T = 273K T = 298K

.O 3.0

4.0 5.0 6.0 Radial distance(Angstroin)

Figure 4.25: Ion pair distribution functions at the different temperatures of 10K(thick solid line), 76K(thick dotted line), 150K(dashed line), 248K(thin solid line), 273K(thin dotted line), and 298K(thick long dashed line).

102

% >

• r

t

. « 4t %

(b)

-V

Figure 4.26: Trajectories of the potassium ions at (a) T=10K, (b) T=76K, (c) T=248K, and (d) T=298K.

103

0.06

T=298K

273K

0.04

Q CO S

0.02

223K .

150K

76K

10K 0.00 10

T(10 '^sec)

Figure 4.27: Mean square displacement (MSD) of potassium ions as a function of time and temperature.

104

i

50 100 150 200 Temperature (K)

250

Figure 4.28: Vibrational amplitude of the potassium ions as function of the ion temper­ature. The dotted line is a linear fit to the data except for the last point.

105

UL < >

T=298K

T=:273K a 0.04

vw\/\Ar^Ay||| S 0.02

T=10K

0 12 3 Frequwieytio" Hz)

T( IO "sec ) Frequency (10 Hz)

Figure 4.29: Normalized velocity autocorrelation function(NVAF) and associated power spectrum for crystalline potassium(T=10K) and liquid metal(T=273K and T=298K). The insert in the T=10K power spectrum is the experimentally deduced phonon density of state at T=9K of a reference [19].

106

% * • • t

Figure 4.30: A 2D projectioa of the electron necklace (open circle) with potassium ions (large filled circle) at T = 298K and T = lOK. The frame represents the simulation cell.

107

1.2

1.0

antiparallel spin

c o 0.8 3 n

parallel spin

(B Q. o • 0.4 o

I

I

0.2

0.0

Radial distance(Angstrom)

Figure 4.31: Partial electron-electron pair correlation functions. The solid lines and dotted lines refer to the crystal at T=10K and the liquid at T=273K, respectively.

108

1.2

1.1

1.0

73 0.9

0.8 T=10K T=150K T=273K

0.7

0.6

Radial distance (Angstrom)

Figure 4.32: Ion-electron pair distribution functions at severaJ temperatures.

109

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

We have shown that a non-local form of the restricted discretized path integral may be

used to define an effective exchange potential for use in molecular dynamics simulations

of quantum particles obeying Fermi statistics. A quantum particle is represented as a

closed necklace of discrete beads. Exchange is described via non-local cross-linking of

the necklaces. First, we have demonstrated that electron plasmas may be simulated

with a satisfactory degree of accurjicy with this method up to metallic densities. We

have noted that the exchange potential appears to introduce correlation in some effec­

tive form. We have applied the first-principles molecular dynamics scheme to the study

of the finite temperature properties of a simple metal. We showed that the model potas­

sium metal undergoes a melting transformation upon heating. The transformation is

characterized thoroughly through calculated thermodynamics quantities, structural and

dynamical properties. The model potassium crystal melts at a temperature significantly

below the experimental melting point. The reasons for this discrepency may be found in

the approximations made to speed up the calculations as for instance the use of a short-

range interatomic potential and an empty core pseudopotential. Comparison between

the calculated low-temperature vibrational spectrum and the experimentally measured

phonon density of states indicates that the strength of the metalic bonds is underes­

timated in the model. Upon melting the ionic motion changes from pure vibration to

diffusive motion. Above the transition temperature, the ion mean square displacement

increases linearly with temperature an unambiguous sign of diffusion. This change in

atomic motion is also supported by the temperature dependency of the Fourier trans­

form of the ion velocity auto-correlation function. The path-integral MD allows us to

110

study the interplay between the atomic and electronic structures. In the crystalline state,

vibrations appear to have an effect on the electron density and result in some electron

localization. Moreover, we find that the electronic structure of the simple metal responds

to the collapse in long range order of the ionic structure by localizing within and at the

edge of the core the ions.

Contrary to many of the current quantum molecular dynamics simulation tech­

niques which rely on the independent particle approximation, the path-integral MD is a

many particle technique and includes the important effects of interactions of electrons

with each other and with the ions. Although path-integral MD a very promising tech­

nique for the study of materials in which electronic and ionic structures are intimately

correlated, the shear computational cost of the algorithm constitutes a barrier to its

application to large systems. At present, the restricted path-integral MD method is

limited by the computation cost of forces derived from the effective exchange potential.

The computational cost is a quadratic function of the number of beads and a cubic func­

tion of the number of isospin electrons. Access to supercomputers can make possible

the simulation of systems with larger numbers of electrons. For larger fermion systems,

one may be able to optimize the calculation by exploitng the short spatial extent of

exchange[47] and dividing the simulation cell into smaller and more tractable subcells.

The quadratic dependency on P due to the non-locality of the exchange potential is a

more serious problem. A local effective exchange potential could lead to a linear de­

pendency on the number of beads. We are currently developing an approximate local

form of the exchange potential that is able to model the exchange interactions at a cost

proportional to P only[61].

Ill

CHAPTER 6

APPENDICES

6.1 APPENDIX A : FREE PARTICLE PROPAGATOR

In this Appendix, we will prove equation (2.16)

We assume that every state is defined by a set of plane waves. The free particle propa­

gator in the position representation can be written as

PQ(ri,rj;e) = (r,| e-'^|r_,>

= J <ip{ri\p){p\e-'P

= j rfp<»-.|p)<p|rj>e-"'^/^'". (6.139)

We used a completeness of momentum states, 1= /dp\p){p\, in Eq. (6.139). If we use

a plane wave basis (r,|p) = ^^"^pT^-exp (ip • r,/^), Eq. (6.139) becomes

4n- /•<» , sin (or;,/ft)

where Vij = Jr,- — rj|. From an integral table, we have

xsin {tx) e~ '^dx = t (6.141)

Thus if we set a = £/2m and t = we have Eq. (6.138).

112

6.2 APPENDIX B : EXCHANGE KINETIC ENERGY ESTIMATOR

In this section, we will ccilculate the exchange kinetic estimator, {KEexch)r which is given

by Eq. (2.86),

(o down P P f \ I (««(£"")) s=up k=l 1=1 /

/dovm . P P 1/5 \

In order to differentiate a determinant, we use the following matrix aigebra: if a .V x N

square matrix A is function of X, we have

N Af det A = 51 Otj Aij = Oij Aij, (6.143)

.=1 j=i

^detA = (6.144) i=l j=l

where a,j is an element of the matrix A and A,j is a cofactor of the element a,j. Thus

if we differentiate det(£''^''') with respect to /?, we have

= EE I (£"•"),. .4, «=1 J=1

^ JL J _ «(0^2 _ ,^(fc) _ J0^2^ U . = E E Hp ((r!" - .yv - (rl" - }]. ,

= iSi E E ((-•!" - -y - (--l" - ••!")') (£"•"). • 'ly- (6.145)

If we apply Eq. (6.143) to the left-hand side of the last term of Eq. (6.145), we can wirte

the equation with a simpler form;

!<!«(£"•')) = <'•'>), (6.146)

113

where is an iV X iV matrix and its element is given by

exp i f i = t

{^n.= (6.147)

and

i(W) = ri') - r<'V - (rl'l -

From Eq. (6.142) and (6.146), the exchange kinetic estimator can be written as

(dovm , P P f iV p„ \ E ^ E E E ««('f;"') • (S.148,

6.3 APPENDIX C : EXCHANGE FORCE CALCULATION

Here we calculate the exchange force. From equation (3.127),

fW _ i. ^ g(£'(**''^))py «("•")) C6 149^ '3EE<,et(£:(».'))EE ( )p» (6-«)

where

= exp{Co((rM)2-(rl'-l)2 + 2r;,''>T(''>-2rJ"l-rM)} (6.150)

with Co = and is a cofactor of a matrix element Using Eq.

(6.150), then we have

5r- '

-rjr^Si,pSk,^ - r^Si^pSk,,) (6.151)

The first two terms of the left-hand side of Eq. (6.151) with (6.149) become

e(f:),lst,2nd Pm njp-Ak},iH,2nd ^ y" (Ji. <5i. J t,exch ^2^2 ^ P "t,q°k,l/)^pq Opg

114

^ ^tkl^ f (k) (n,k) ^ ^2^2 I ^ ' '' ^l'' '"'"

= 0. (6.152)

In Eq. (6.152), we used the relation det(£'^^''*') = det(£'^'^'^)) and the matrix algebra

Eq. (6.143). From the third and the fifth terms , and the fourth and sixth terms of

equation (6.151), we have

P P a+ ip» S' N'

Jt,exch ft2fi2 2^ 2^ (1&t( ^ f P t.P"k,ii)^pq Opq

Pm ^ &tJP' _-(>'Up(i',u)Mk,u) , /32^2 det(£:(*''')) '' ' (b-15J)

and similarly

P P fl+ /P- 'V' N' Ak)Ath,6th ^ fm ^ _-(/x)x. X,

•'i.excA jg2 2 Z-f 2^ \ 2-f P P t-P"k,u)£'pq "p, a=l "=1 ^ ' p=l 7=1

P™ - - J:

^p=l p=l

£(r('') - rS''>)£'Jr'*^B<f (6.154) M=1 p=l

Form the above equations, the exchange force at the A:th bead of the ith electron becomes

fl'l^ = If S d^i£i) } (6-155)

where the elements of matrix pj and (jJ"'*' are defined as

(rl"l - r|"'){E('-»))„ if p=i

;32ft2 2^ M=i det(f;(''.*))

Pm p

Kk/p-^2ft2 2-r

/t=l det(E(»-'>)

Pm P »lklP-

= •(

, (£"'"')p, if P¥^i-

and

(£;('''%, if q=i

(6.156)

(G1»'")„ = (£<'-*l)„ i f q ^ i

(6.157)

115

REFERENCES

[1] A. Alavi aad D. Frenkel, J. Chem. Phys. 97, 9249 (1992).

[2] A. Alavi, J. Kohanoff, M. Parrinello, and D. Frenkel, Phys. Rev. Lett. 73, 2599

(1994).

[3] C.B Alcock, J. of Physical and Chemical Reference Data, 23, 385 (1994).

[4] B.J. Alder and J.E. Wainwright, J. Chem. Phys. 27, 1208 (1957),

[5] B.J. Alder and J.E. Wainwright, J. Chem. Phys. 31, 456 (1958), and ibid, 33, 1439

(1960).

[6] H.C. .A.ndersen, J. Chem. Phys. 72, 2384 (1980).

[7] J.E. Anderson, J. Chem. Phys. 63, 1499 (1975), ibid 65, 4121 (1976), and ibid 73,

3897 (1980).

[8] P.E. Blochl and M. Parrinello, Phys. Rev. B 45, 9413 (1992).

[9] S.G. Brush, H.L. Sahlin, and E. Teller, J. Chem. Phys. 45, 2102 (1966).

[10] D.J.E. Cailaway and A. Rahman, Phys. Rev. Lett. 49, 613 (1982).

[11] R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).

[12] D.M. Ceperley, Phys. Rev. B 18, 3126 (1978).

[13] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1980).

[14] D.M. Ceperley, J. Stat. Phys. 63, 1237 (1991).

[15] D.M. Ceperley, Phys. Rev. Lett. 69, 331 (1992).

116

[16] D.M. Ceperiey, Phys. Rev. Lett. 73, 2145 (1994).

[17] D.M. Ceperiey, Rev. Mod. Phys. 67, 279 (1995).

[18] D. Chandler and P.G. Wolynes, J. Chem. Phys. 74, 4078 (1981).

[19] R.A. Cowley, A.D.B. Woods, and G Dolling, Phys. Rev. 150, 487 (1966).

[20] R.G. Dendrea, N.W. AshcrofFt, and A.E. Carlsson, Phys.Rev. B 34, 2097 (1986).

[21] P.P. Ewald, Ann. Physik. 64, 253 (1921).

[22] R.P. Feynman, Phys. Rev. 76, 769 (1949), ibid 90, 1116 (1953), ibid 91, 1291

(1953), and ibidQl, 1301 (1953).

[23] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-

Hil, New York, 1965).

[24] R.P. Feynman, Statistical mechanics, (Benjamin, New York, 1972).

[25] L.D. Fosdick and H. F. Jordan, Phys. Rev. 143, 58 (1966), ibid 171, 128 (1968).

[26] W. Freyland, Phys. Rev. B 20, 5104 (1979).

[27] F.G. Fumi and M.P. Tosi, J, Phys. Chem. Solids, 25, 31 (1964).

[28] M.Gell-Mann and K. Bruekner, Phys. Rev. 106, 364 (1957).

[29] D.A. Gibson and E.A. Carter, J. Phys. Chem. 97, 13429 (1993).

[30] M.J. Gillan, in Computer Modelling of Fluids, Polymers and Solids, C.R.A. Catlow

et al. Eds. (Dordrecht, Kluwer Academic, 1990), p.l55.

[31] W. Gropp, E. Eusk, and A. Skjellum, Using MPI: Portable Parallel Programming

xnith Message-Passing Interface, (MIT Press, Cambridge, 1994).

[32] M. Haile and S. Gupta, J. Chem. Phys. 79, (1983).

[33] R.W. Hall and B.J. Berne, J. Chem. Phys. 81, 3641 (1984).

117

[34] R.W. Hall, J. Chem. Phys. 89, 4212 (1988).

[35] R.W. Hall, J. Chetn. Phys. 91, 1926 (1989).

[36] R.W. HaU, J. Chem. Phys. 93, 5628 (1989).

[37] R.W. Hall, J. Chem. Phys. 93, 8211 (1990).

[38] W.A. Harrison, Pseudopotentials in the Theory of Metals (New York : Benjamin)

1966.

[39] M.F. Herman, E.J. Bruskin, and B.J. Berne, j. Chem. Phys. 76, 5150 (1982),

[40] D.W. Hermann, Computer Simulation Method in Theoretical Physics, (Springer-

Verlag, Berlin, 1986).

[41] W.G. Hoover, A.J.C. Ladd, and B. Moran, Phys Rev. Lett. 48, 1818 (1982).

[42] C. Kittel, Elementary Statistical Physics, (John Wiley & Sons, New York, 1967).

[43] C. Kittel, Introduction to Solid State Physics ed. (Wiley, New York, 1992).

[44] H. Kleinert, Path Integral Quantum Mechanics, Statistics, and Polymer Physics

(Singapore : World Scientific, 1990).

[45] A.J.C. Ladd and W.G. Hoover, Phys. Rev. B 28, 1756 (1983).

[46] C.Y. Lee and P.A. Deymier, Solid State Comm. 102, 653 (1997).

[47] X.P. Li, R.W. Numes, and D. Vanderbilt, Phys. Rev. B 47, 10891 (1993).

[48] P. Linse and H.C. Anderson, J. Chem. Phys. 85, 3027 (1986).

[49] G.J. Martyna, M.L. Klein, and M. Tuckerman, J. Chem. Phys. 97, 2635 (1992).

[50] D. Marx and M. Parrinello, Z. fur Physik B 95, 143 (1994).

[51] M.A. McQuarrie, Statistical Mechanics, (Harper and Row, New York, 1976).

[52] N. Metropolis and S. Ulam, J. Am. Stat, Assoc. 44, 335 (1949).

118

[53] A. Muramatsu et al. Int. J. Mod. Phys. C3, 185 (1992).

[54] J.W. Negele and H. Orland, Quantum Many-Particle Systems, (Addison-Wesley,

New York, 1988).

[55] B.R.A. Nijboer and F.W. De Wette, Physica, 23, 309 (1957).

[56] S. Nose, J. Chem. Phys. 81, 511 (1984).

[57] S. Nose, J. Chem. Phys. 81, 511 (1984).

[58] Ki-Dong Oh and P.A. Deymier, Phys. Rev. Lett. 81, 3104 (1998).

[59] Ki-Dong Oh and P.A. Deymier, Phys. Rev. B 58, 7577 (1998).

[60] Ki-Dong Oh and P.A. Deymier, Phys. Rev. B, in press.

[61] Ki-Dong Oh and P.A. Deymier, in preparation.

[62] M. Parrineilo and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980).

[63] M. Parrineilo and A. Rahman, J. Appl. Phys. 52, 7182 (1981).

[64] M. Parrineilo and A. Rahman, J. Chem. Phys. 80, 860 (1984).

[65] M. Parrineilo, Solid State Comm. 102, 107 (1997).

[66] F. Perrot and M.W.C. Dharraa-wardana, Phys. Rev. A 30, 2619 (1984).

[67] E.L. Pollock and D.M. Ceperley, Phys. Rev. B 30, 2555 (1985), ibid 36, 8343

(1987).

[68] W.H. Press et ai. Numerical Recipes in C, (Cambridge Univ. Press, Cambridge,

1988).

[69] G-X Qian, M. Weinert, G.W. Fernando, and J.W. Davenport, Phys. Rev. Lett.

64, 1146 (1990).

[70] A. Rahman, Phys. Rev. 159, 98 (1967).

119

[71] M.J.L. Sangster and M. Dixon, Adv. in Phys. 25, 247-342 (1976).

[72] M.J.L. Sangster and R.M. Atwood, J. Phys. Chem. C 11, 1541 (1978).

[73] G. Senger, M.L. Ristig, C.E. Campbell, and J.W. Clark, Annals of Physics 218,

160 (1992).

[74] A. Selloni, P. Carnevali, R. Car, and M. Parrinello, Phys. Rev. Lett. 59, 823 (1987).

[75] B. Simon, Funtional Integrals and Quantum Mechanics, (Academic Press, New

York, 1979).

[76] C.J. Smithell, Metals Reference Handbook, 7th ed. (Butterworth, London, 1992).

[77] R.M. Stratt, Annu. Rev. Phys. Chem. 41, 175 (1990).

[78] J. Thalhaiier, Phys. Fluids B 4, 2044 (1992).

[79] S. Tanaka, S. Mitake, and S. Ichimaru, Phys. Rev. .A. 32, 1896 (1985).

[80] E. Teller, J. Chem. Phys. 21, 1087 (1953).

[81] M.P. Tosi and F.G. Fumi, J. Phys. Chem. Solids, 25, 45 (1964).

[82] H. Trotter, Proc. Am. Math. Soc, 10, 545 (1959).

[83] M.E. Tuckerman, B.J. Berne, and A. Rossi, J. Chem. Phys. 94, 1465 (1991).

[84] M.E. Tuckerman, B.J. Berne, and G.J. Martyna, J. Chem. Phys. 97, 1990 (1992).

[85] M.E. Tuckerman, B.J. Berne, G.J. Martyna, and M.K. Klein, J. Chem. Phys. 99,

2796 (1993).

[86] L. Verlet, Phys. Rev. bf 159, 98 (1967).

[87] S. Ulajn, .4 Collection of Mathematical Problems, (Intersdence, New York, 1960).

120

[88] W.W. Warren, Jr., in The Metallic and Nonmetallic States of Matter, P.P. Edwards

and C.N.R. Rao, editors, (Taylor and Francis, London, 1985).

[89] Y. Waseda, in The Structure of Non-crystalline Materials (McGraw Hill, New

York) 1980.

[90] S.R. White, Phys. Rev. B41, 9031 (1990).

[91] S.R. White, Phys. Rev. Lett- 69, 2863 (1992).

[92] W.W Wood and F.R. Parker, J. Chem. Phys. 27, 720 (1957).

[93] L.V. Woodcock, Chem. Phys. Lett. 10, 257 (1971).

[94] L.V. Woodcock and Singer, Trans. Fraday Soc. 67, 12 (1971).

[95] S. Zhang, J, Carlson, and J.E. Gubernatis, Phys. Rev. B 55, 7464 (1997).

IMAGE EVALUATION TEST TARGET (QA-3)

150mm

IM/IGE. Inc 1653 East Main Street Rochester, NY 14609 USA Phone: 716/482-0300 Fax; 716/288-5989